Fractional-order linear time invariant swarm systems: asymptotic swarm stability and time response analysis

Cent Eur J Phys 11(6) 013 845-854 DOI: 10478/s11534-013-074-5 Central European Journal of Physics Fractional-order linear time invariant swarm systems: asymptotic swarm stability and time response analysis Research Article Mojtaba Naderi Soorki 1,, Mohammad Saleh Tavazoei 1, 1 Electrical Engineering Department, Sharif University of Technology, Azadi Avenue, Tehran, Iran Received 7 February 013; accepted 6 June 013 Abstract: PACS (008): 4530+s Keywords: This paper deals with fractional-order linear time invariant swarm systems Necessary and sufficient conditions for asymptotic swarm stability of these systems are presented Also, based on a time response analysis the speed of convergence in an asymptotically swarm stable fractional-order linear time invariant swarm system is investigated and compared with that of its integer-order counterpart Numerical simulation results are presented to show the effectiveness of the paper results fractional-order linear time invariant swarm systems asymptotic swarm stability time response analysis Versita sp z oo 1 Introduction In recent years, multi-agent autonomous systems imitating natural swarms have attracted much interests [1, ] Some real-world examples of swarms include mass movement of birds, schools of fish, and crowds of people [3] Swarm systems have many applications in various areas, such as physics, biology, robotic and engineering [4 6] Recently emerging applications such as formation control and distributed coordination of multi robot systems have increased the interest of engineers in swarms The stability analysis of swarm systems is more challenging due to the difference between these systems and isolated systems Paper [7] has investigated the consensus for first- E-mail: mojtabanaderi@eesharifedu E-mail: tavazoei@sharifedu (Corresponding author) order continuous-time swarm systems In paper [8], it has been shown that the existence of an interaction graph including a spanning tree is necessary for asymptotic swarm stability Besides these, there are also many studies related to the stability of swarm systems [9, 10] Paper [11] has presented sufficient conditions for consensus of highorder linear time-invariant swarm systems Linear time invariant swarm systems with specific structures have been studied in [1] and [13] Recently, paper [14] has presented necessary and sufficient conditions for swarm stability of high-order linear time invariant swarm systems The most of these stability studies of swarm systems have been limited to integer-order cases, even though fractional order control systems have attracted increasing attention in the past few decades [15 18] As a related work, the distributed coordination of networked fractional-order systems over a directed interaction graph has been studied in [19] In paper [19], sufficient conditions on the interac- 845

Fractional-order linear time invariant swarm systems: asymptotic swarm stability and time response analysis tion graph and the fractional order have been found such that the coordination is achieved This paper presents necessary and sufficient conditions for asymptotic swarm stability of fractional-order linear time invariant swarm systems Also, the time response of an asymptotically swarm stable fractional-order linear time invariant swarm system is investigated and compared with that of its integer-order counterpart To the best of the authors knowledge, this paper is the first research work that introduces fractional-order linear time invariant swarm systems and investigates the properties of these systems In fact, the study of swarm stability and time response analysis are new challenges in the field of swarm systems which are addressed in this paper for fractionalorder linear time invariant swarm systems This paper is organized as follows A model for a fractional-order linear time invariant swarm system is presented in Section Moreover, the asymptotic swarm stability of such a model is investigated in this section Section 3 is devoted to the time response analysis of asymptotically swarm stable fractional-order linear time invariant swarm systems Also, numerical simulation results are presented in Section 4 Finally, conclusions are given in Section 5 Fractional-order swarm systems and their stability analysis Fractional-order linear time invariant swarm systems and their stability analysis are addressed in this section These systems are modelled as having N agents whose dynamics are described by pseudo state space models having inner dimension d It is assumed that the pseudo state of the ith agent is denoted by x i = [x i1,, x id ] T R d The communication among agents is specified by a weighted directed graph G of order N such that each agent corresponds to a vertex Also, the arc weight of graph G between ith and jth agents is specified by the value w ij 0 This value can be considered as a measure for the strength of the information link [14] Graph G is specified by its adjacency matrix which is denoted by W w 11 w 1N G : W = w N1 w NN Let the dynamics of each agent be described by pseudo state space models D α t x i = Ax i + F N w ij (x j x i ), i {1,,, N} (1) j=1 where A R d d, F R d d, and Dt α denotes the Caputo fractional derivative operator defined as follows [1] D α t f(t) = { 1 t Γ(n α) 0 f (n) (τ) dτ, n 1 < α < n N (t τ) α n+1 f (n) (t), α = n N In the above definition, n is the first integer larger than or equal to α It is worth mentioning that the considered swarm system is the fractionalized version of that introduced in [14] Now, consider the following definition Definition 1 Asymptotic swarm stability [14]: A linear time invariant swarm system is said to be asymptotically swarm stable if it achieves the full state consensus, ie for each ε > 0 there is t > 0 such that x i (t) x j (t) < ε for t > t (i, j {1,,, N}) If the pseudo state vector of agents is defined by x = [x T 1,, xt N ]T, the system dynamics in (1) can be expressed as D α t x = (I N A L F)x () where L = L(G) is the Laplacian matrix of the graph G [0] The following lemma reveals a property of the Laplacian matrix of a graph Lemma 1 The Laplacian matrix of any graph always has a zero eigenvalue Also, the Laplacian matrix of a directed graph has exactly one zero eigenvalue λ 1 = 0 with the corresponding eigenvector [1, 1,, 1] T if and only if this graph includes a spanning tree In such a case, all of the other eigenvalues λ,, λ N are placed in the open right half plane [7, 8] By considering the eigenvalues of L as λ 1 = 0, λ,, λ N C, the Jordan canonical form of L can be written as 0 0 0 0 0 λ 0 J = 0 0 0 0 0 λ N where may either be 1 or 0 By assuming T LT 1 = J and defining x = (T I d )x, () is rewritten as D α t x = (I N A J F) x (3) 846

Mojtaba Naderi Soorki, Mohammad Saleh Tavazoei where matrix (I N A J F) is in the form A 0 0 0 0 A λ F 0 (I N A J F) = 0 0 0 0 0 A λ N F (4) In (4), each represents a matrix in R d d that may either equal F or 0 (For more details, see [14]) According to (4), (3) can be decoupled into pseudo state space equations having one of the following forms or D α t x i = (A λ i F) x i (5) D α t x i = (A λ i F) x i F x i+1 (6) Before deriving the main result, we need the following two lemmas that are similar to what was expressed in [19] (In fact, Lemmas and 3 are vector forms of Lemmas and 3 of [19], which investigate the time response properties of systems (5) and (6) in the scalar case) Lemma Let x i (t) be the solution of (5) following properties: This solution has the 1 If α (0, θ i π ), where θ i = min arg(eig(a λ i F)) then lim t x i (t) = 0 If α (0, 1] and λ i = 0, then x i (t) = E α,1 (At α ) x i (0) 1 3 If α (1, ) and λ i = 0, then x i (t) = E α,1 (At α ) x i (0) + te α, (At α ) x i (0) Proof By the stability theorem of fractional-order systems described by pseudo state space models [3], it can be easily shown that if α (0, θ i ), where θ π i = min arg(eig(a λ i F)), then system (5) will be an asymptotically stable system, and consequently lim t x i (t) = 0 Also, properties and 3 can be respectively concluded from [[4]: Eq (347)] and [[5]: Eq (16)] 1 E α,β () denotes the two-parameter Mittag-Leffler function [1] Lemma 3 Let x i+1 (t) be a continuous function satisfying condition lim t x i+1 (t) = 0 In this case, (6) yields in lim t x i (t) = 0, provided that α (0, θ i ), where θ π i = min arg(eig(a λ i F)) Proof By taking the Laplace transform from the both sides of (5), it is deduced that L{ x i (t)} = [s α I A+λ i F] 1 (s α 1 x i (0) FL{ x i+1 }), α (0, 1] (7) L{ x i (t)} =[s α I A + λ i F] 1 (s α 1 x i (0) + s α 1 xi (0) FL{ x i+1 }), α (1, ) (8) If α (0, θ i ), where θ π i = min arg(eig(a λ i F)), the roots of equation det(s α I A + λ i F) = 0 are placed in the left half plane Therefore, in such a case the final value theorem can be used to obtain lim t x i (t) from equalities (7) and (8) By applying this theorem, we have lim x i(t) = lim s[s α I A + λ i F] 1 (s α 1 x i (0) FL{ x i+1 }) t s 0 lim x i(t) = lim s[s α I A + λ i F] 1 (s α 1 x i (0) + s α 1 x i (0) t s 0 FL{ x i+1 }) (10) By considering (9), (10), and assumption lim t x i+1 (t) = 0 which results in lim t L{ x i+1 (t)} = 0, it is concluded that lim t x i (t) = 0 Now, consider the following theorem as the main result of this section Theorem Let α (0, θ ) where θ = min π i=1,,n θ i, θ i = min arg(eig(a λ i F)), and λ 1 = 0,and λ,, λ N C are eigenvalue of the Laplacian matrix L with the Jordan form J = T LT 1 In this case, the swarm system (1) which does not satisfy condition arg(eiga) > α π is asymptotically swarm stable if and only if the graph topology G has a spanning tree In addition, in the steady state the pseudo states of the agents of the asymptotically swarm stable system are obtained as follows x(t) = (T I d ) 1 x(t) lim x(t) = [E α,1(at α ) x 1 (0), 0,, 0] T, α (0, 1] t (9) lim x(t) = [E α,1(at α ) x 1 (0) + te α, (At α ) x 1 (0), 0,, 0] T, t α (1, θ π ) (11) 847

Fractional-order linear time invariant swarm systems: asymptotic swarm stability and time response analysis Proof If graph G has a directed spanning tree, from Lemma 1 it deduced that L has a simple zero eigenvalue, and other eigenvalues L of have positive real parts Since λ 1 = 0 is a simple zero eigenvalue for L, from Properties and 3 of Lemma it is concluded that x 1 (t) = E α,1 (At α ) x 1 (0) for the case α (0, 1] and x 1 (t) = E α,1 (At α ) x 1 (0) + te α, (At α ) x 1 (0) for the case α (1, ) On the other hand, if condition α (0, θ ) is satisfied, Property 1 of Lemma and Lemma 3 yield in π lim 0 x i (t) = 0 for i 1 Considering these points, the asymptotic relations (11) are derived Hence, if graph G has a directed spanning tree and α (0, θ ), we have π E α,1 (At α ) x 1 (0) lim x(t) = (T I d) 1 x(t) 0 = (T 1 I d ) t 0 ˆt 11 I d E α,1 (At α ) x 1 (0) = (1) ˆt 11 I d E α,1 (At α ) x 1 (0) for the case α (0, 1], and lim x(t) = (T I d) 1 x(t) t E α,1 (At α ) x 1 (0) + te α, (At α ) x 1 (0) 0 = (T 1 I d ) 0 ˆt 11 I d E α,1 (At α ) x 1 (0) + te α, (At α ) x 1 (0) = ˆt 11 I d E α,1 (At α ) x 1 (0) + te α, (At α ) x 1 (0) (13) for the case α (0, θ ), where [ˆt π 11,,ˆt 11 ] T is the first column of matrix T 1 (Note that from Lemma 1 we know that this vector is a coefficient of [1, 1,, 1] T ) According to (1) and (13), x 1 (t) = = x N (t) when t which results in the asymptotic swarm stability of the considered swarm system If the interaction graph G does not include a spanning tree and condition arg(eiga) > α π is not satisfied, by the same approach presented in the proof of Theorem 1 in [14], it can be shown that the considered swarm system cannot be asymptotically swarm stable For the case α = 1 (integer-order case), similar that presented in Theorem of [14], Theorem 1 reveals that the considered swarm system with non-hurwitz matrix A is asymptotically swarm stable if and only if matrices A λ i F for i =,, N are Hurwitz and the graph topology G has a spanning tree 3 Time response analysis Theorem 1 in the previous section has specified the pseudo states of agents in the steady state Based on this theorem, in this section some analyses are presented on the time response of an asymptotically swarm stable fractional-order linear time invariant swarm system with an order in the range (0, 1) Also, the obtained results are compared with the results of the integer-order case, ie the case α = 1 Note that the Jordan canonical form of structure (I N A J F) in (3) can be expressed as follows: γ 1 0 0 0 0 γ 0 φ B 1 (I N A J F)B = (14) 0 0 0 0 0 In (14), γ i (i = 1,, Nd) denotes the eigenvalues of the matrices A λ i F where λ 1 = 0, and λ,, λ N C are eigenvalues of the Laplacian matrix L and each may either be 0 or 1 For simplicity, in the rest of this section we assume that the eigenvalues γ i (i = 1,, Nd) are distinct and nonzero According to the structure of (I N A J F), matrix B has the following form S d d 0 B = d d 0 d d (15) where S d d is the similarity transformation matrix used to diagonalize matrix A According to Theorem 1, for an asymptotically swarm stable fractional-order linear time invariant swarm system the eigenvalues of matrices A λ i F for i =,, N are in the stable region, ie arg(eig(a λ i F)) > α π By transformation z = B 1 x, equation (3) can be rewritten as D α t z(t) = φz(t) (16) Since it is assumed that the eigenvalues γ i (i = 1,, Nd) are distinct, the solution of (16) is obtained as [] z(t) ={c 1 Í 1 E α,1 (γ 1 t α ) + c Í E α,1 (γ t α ) + + c Nd INd E α,1 ( t α )} (17) where Íi(i = 1,, Nd) are Nd 1 vectors in which the ith element is 1 and the other elements are zero Also, c i (i = 848

Mojtaba Naderi Soorki, Mohammad Saleh Tavazoei 1,, Nd) are fixed values obtained based on the initial conditions Finally, according to relations x = (T I d )x and x = Bz(t), agents responses are obtained as follows By defining M = (T 1 I d )B R Nd Nd and considering the structure of matrix B in (15), x(t) = (T I d ) 1 x = (T 1 I d )Bz(t) (18) ˆt 11 I d Ŝ d d ˆm d d M = (T 1 ˆt I d )B = 11 I d 0 d d ˆm = d d ˆt 11 I d 0 d d ˆm d d (19) where ˆm d d = ˆt 11 I d S d d (19) can be expressed as follows x 1 (t) m 11 m 1Nd c 1 E α,1 (γ 1 t α ) x(t) = = x N (t) m Nd1 m NdNd c Nd E α,1 ( t α ) m 11 c 1 E α,1 (γ 1 t α ) + + m 1,Nd c Nd E α,1 ( t α ) m d1 c 1 E α,1 (γ 1 t α ) + + m d,nd c Nd E α,1 ( t α ) = m Nd d+1,1 c 1 E α,1 (γ 1 t α ) + + m Nd d+1,nd c Nd E α,1 ( t α ) m Nd,1 c 1 E α,1 (γ 1 t α ) + + m Nd,Nd c Nd E α,1 ( t α ) (0) As equation (0) shows, the time response of each agent can be stated in terms of Mittag-Leffler functions Some parts of the solution related to the eigenvalues of matrices A λ i F(i =,, N) tend to zero as t (These eigenvalues satisfy condition arg(γ i ) > α π ) Also, the terms related to the eigenvalues of matrix A not satisfying condition arg(γ i ) > α π do not converge to zero Now, using Theorem 3 of [19], in the following theorem the speed of convergence in an asymptotically swarm stable fractional-order swarm system is compared with that of its integer-order counterpart for sufficiently small times Theorem 31 There is T > 0 such that the elements of difference vector between pseudo states of the ith agent and the jth agent (ie vector x i (t) x j (t)) in an asymptotically swarm stable fractional-order swarm system with an order between 0 and 1 (ie 0 < α < 1 ) decrease faster than those of its integer-order counterpart for t (0, T ) Proof Suppose that γ l for l = 1,, d are the eigenvalues of matrix A, and γ l for l = d+1,, Nd are eigenvalues of matrices A λ i F(i =,, N) In an asymptotically swarm stable fractional-order swarm system, we know that arg(γ l ) > α π for l = d + 1,, Nd (Theorem 1) According to (0), the difference of vectors x i (t) and x j (t) can be expressed as follow 849

Fractional-order linear time invariant swarm systems: asymptotic swarm stability and time response analysis m id d+1,1 c 1 E α,1 (γ 1 t α ) + + m id d+1,nd c Nd E α,1 ( t α ) x i (t) x j (t) = m id,1 c 1 E α,1 (γ 1 t α ) + + m id,nd c Nd E α,1 ( t α ) m jd d+1,1 c 1 E α,1 (γ 1 t α ) + + m jd d+1,nd c Nd E α,1 ( t α ) m jd,1 c 1 E α,1 (γ 1 t α ) + + m jd,nd c Nd E α,1 ( t α ) m id d+1,1 m jd d+1,1 m id d+1,d m jd d+1,d c 1 E α,1 (γ 1 t α ) = m id,1 m jd,1 m id,d m jd,d c d E α,1 (γ d t α ) m id d+1,d+1 m jd d+1,d+1 m id d+1,nd m jd d+1,nd c d+1 E α,1 ( t α ) + m id,d+1 m jd,d+1 m id,nd m jd,nd c Nd E α,1 ( t α ) (1) According to (19), m id d+1,1 m id d+1,d = m id,1 m id,d m jd d+1,1 m jd d+1,d = ˆm d d () m jd,1 m jd,d Hence, (1) can be written as m id d+1,d+1 m jd d+1,d+1 m id d+1,nd m jd d+1,nd c d+1 E α,1 ( t α ) x i (t) x j (t) = (3) m id,d+1 m jd,d+1 m id,nd m jd,nd c Nd E α,1 ( t α ) If α = 1 (the integer-order case), (3) results in m id d+1,d+1 m jd d+1,d+1 m id d+1,nd m jd d+1,nd c d+1 e t x i (t) x j (t) = (4) m id,d+1 m jd,d+1 m id,nd m jd,nd c Nd e t By comparison the convergence rate of functions E α,1 (γ i t α ) and e γ it for sufficiently small t (See Theorem 3 of [19]), from (3) and (4) one can conclude that there exists T > 0 such that the elements of difference vector x i (t) x j (t) in an asymptotically swarm stable fractional-order swarm system with 0 < α < 1 decrease faster than those of its integer-order counterpart for t (0, T ) As an important result, Theorem reveals that in asymptotically swarm stable fractional-order swarm systems the convergence rate of agents in the initial moments of motion is higher than that of their integer-order counterparts This means that in the fractional-order case the convergence speed is fast in the beginning of motion, ie when the initial distance between agents may be large Now, we want to study the speed of convergence for the sufficiently large times To achieve this goal, first consider the following lemma from [1] Lemma 31 If 0 < α < and µ is an arbitrary real number such that πα < µ < min{π, πα}, then for each γ i C satisfying condition µ arg(γ i t α ) π, we have E α,1 (γ i t α ) ( t α ) (γ i Γ(1 α)) (5) 850

Mojtaba Naderi Soorki, Mohammad Saleh Tavazoei as t The following theorem reveals the convergence rate in an asymptotically swarm stable fractional-order swarm system for sufficiently large times Theorem 3 In an asymptotically swarm stable fractional-order swarm system in the form (1) with the order α (0, 1), the average distance between agents decreases like as t α when t if the eigenvalues of matrices A λ i F for i = 1,, N are different and nonzero Proof Let γ 1,, γ d be the eigenvalues of matrix A and,, be the eigenvalues of matrices A λ i F for i = d + 1,, Nd In an asymptotically swarm stable fractional-order swarm system, arg(γ i ) > α π for i = d + 1,, Nd Hence, according to (1) and Lemma 31 one can concluded that m 11 c 1 E α,1 (γ 1 t α ) + + m 1,d c d E α,1 (γ d t α ) x(t) m Nd,1 c 1 E α,1 (γ 1 t α ) + + m Nd,d c d E α,1 (γ d t α ) m 1,d+1 c d+1 + + m 1,Ndc Nd + t α Γ(1 α) m Nd,d+1 c d+1 + + m Nd,Ndc Nd (6) when t Vector x(t) R Nd in (6) can be expressed as x1 1 (t) m 11 c 1 E α,1 (γ 1 t α ) + + m 1,d c d E α,1 (γ d t α ) x 1 (t) x x(t) = d 1 m d1 c 1 E α,1 (γ 1 t α ) + + m d,d c d E α,1 (γ d t α ) = x N (t) x1 N m Nd d+1,1 c 1 E α,1 (γ 1 t α ) + + m Nd d+1,d c d E α,1 (γ d t α ) xd N m Nd,1 c 1 E α,1 (γ 1 t α ) + + m Nd,d c d E α,1 (γ d t α ) m 1,d+1 c d+1 + + m 1,Ndc Nd + t α Γ(1 α) m d,d+1 c d+1 m Nd d+1,d+1 c d+1 m Nd,d+1 c d+1 + + m d,ndc Nd + + m Nd d+1,ndc Nd + + m Nd,Ndc Nd (7) Let the average distance between agents be defined by h(t) = N 1 (N (N 1)) N k=1 i=k+1 (x k 1 (t) xi 1 (t)) + + (x k d (t) xi d (t)) (8) From (19), m 11 m 1,d ˆm d d = = = m d,1 m d,d m Nd d+1,1 m Nd d+1,d (9) m Nd,1 m Nd,d According to (7) (9), h(t) pt α Γ(1 α)n (N 1) (30) 851

Fractional-order linear time invariant swarm systems: asymptotic swarm stability and time response analysis where the constant p is given by p = N k=1 i=k+1 N Nd j=d+1 c j γ j (m dk (d 1),j m di (d 1),j )) + + ( Nd j=d+1 c j (m dk,j m di,j ) γ j (31) Theorem 3 reveals that for sufficiently large times the speed of convergence in an asymptotically swarm stable fractional-order swarm system is less than that of its integer-order counterpart 4 Numerical simulations In this section, the effectiveness of the presented analysis is shown by a numerical example (Numerical simulations of this section have been carried out by using the Adams-type predictor-corrector method which has been introduced in [6] for solving fractional-order differential equations) To this end, consider the following fractionalorder linear time invariant swarm system D α t x i = Ax i + F 5 w ij (x j x i ), i {1,, 5} (3) j=1 which has 5 agents and the pseudo state vector of each agent is denoted by x i = [x i1, x i ] T R Graph G a shown in Figure 1 is used to express the communication among agents with 0 0 0 0 05 075 0 0 0 045 W = 0 03 0 0 0 06 0 0 0 0 0 0 0 03 0 The eigenvalues of the Laplacian matrix for this graph are λ(g a ) = {0, 1, 03, 07 ± 0374i} Assume that α = 08 and matrices A and F are given by [ ] [ ] 0 03 7 9 A =, A = 5 8 6 In this case, the eigenvalues of matrix A are 1 ± 071i that are not in the stable region {s C arg(eig(s)) > Figure 1 Graph G a which includes a spanning tree α π }, whereas the eigenvalues of A λ if(i =,, 5) are 095 ± 457i, 686 ± 588i, 04 ± 1074i and 68 ± 163i are in the aforementioned stable region According to Theorem 1, the swarm system in question is asymptotically swarm stable Figure shows the trajectories of the agents which confirms the consensus Also, Figure 3 shows the average distance between agents, ie function h(t) = 1 4 5 10 k=1 i=k+1 (xk1 (t) x i1 (t)) + (x k (t) x i (t)) To verify the results of Theorem 3, the diagram log(h(t)) versus log(t) has been plotted in Figure 4 For large enough times, this diagram specifies a line with slope α = 08 which confirms the results of Theorem 3 5 Conclusion In this paper, fractional-order linear time invariant swarm systems were introduced and studied in the aspect of swarm stability and time response analysis The main contributions of the paper can be summarized as follows: Finding a closed-form expression for time response of fractional-order linear time invariant swarm systems (Equation (0)) Providing an asymptotic swarm stability analysis for fractional-order linear time invariant swarm systems (Theorem 1) 85

Figure 1 Graph which includes a spanning tree Mojtaba Naderi Soorki, Mohammad Saleh Tavazoei trajectory of agents 10 0 y -10-0 -30-40 -50-60 -70-80 -40-35 -30-5 -0-15 -10-5 0 5 10 x Figure 4 log(h) versus log(t) Figure Trajectories Figure Trajectories of agents in the swarm system considered in of agents in the swarm system considered in Section 4 ( [0,7] ) Section 4 (t [0, 7]) Acknowledgments The authors thank the Research Council of Sharif University of Technology for supporting this work References Figure 3 Average distance between agents Comparing the speed of convergence in an asymptotically swarm stable fractional-order linear time invariant swarm system with that of its integerorder counterpart for sufficiently small and large times (Theorems 31 and 3) The obtained results in this paper can be considered as a basis for further research works on fractional-order swarm systems [1] J K Parrish, Science 84, 99 (1999) [] J Buhl, et al, Science 31, 140 (006) [3] D Helbing, I Farkas, T Vicsek, Nature 407, 487 (000) [4] T Vicsek, A Czirok, E Ben-Jacob, I Cohen, O Shochet, Phys Rev Lett 75, 16 (1995) [5] R Olfati-Saber, IEEE T Automat Contr 51, 401 (006) [6] F Xiao, L Wang, J Chen, Y Gao, Automatica 45, 605 (009) [7] R Olfati-Saber, RM Murray, IEEE T Automat Contr 49, 150 (004) [8] W Ren, RW Beard, IEEE T Automat Contr 50, 655 (005) [9] W Li, IEEE T Syst Man Cy B 38, 1084 (008) [10] A Pant, P Seiler, JK Hedrick, IEEE T Automat Contr 47, 403 (00) [11] J Wang, D Cheng, X Hu, Asian J Control 10, 144 (008) [1] W Ren, KL Moore, Y Chen, J Dyn Systy T ASME 19, 678 (007) [13] P Wieland, J Kim, H Scheu, F Allgower, On consensus in multi-agent systems with linear high-order agents Proc IFAC World Congress, Seoul, Korea 17, 1541 (008) 853

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