Linear Algebra and its Applications

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1 Linear Algebra and its Applications 431 (9) Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: On the general consensus protocol of multi-agent systems with double-integrator dynamics Jiandong Zhu a,, Yu-Ping Tian b, Jing Kuang a a School of Mathematics and Computer Science, Nanjing Normal University, Nanjing 197, PR China b School of Automation, Southeast University, Nanjing 196, PR China A R T I C L E I N F O A B S T R A C T Article history: Received 9 September 8 Accepted 1 March 9 Available online April 9 Submitted by H. Bart Keywords: Multi-agent systems Consensus Double-integrator dynamics In this paper, a general consensus protocol is considered for multiagent systems with double-integrator dynamics. The advantage of this protocol is that different consensus dynamics including linear, periodic and positive exponential dynamics can be realized by choosing different gains. Necessary and sufficient conditions for solving the consensus problem with the considered general protocol are obtained, namely, all the gains realizing the consensus can be described. The design method of the consensus protocol is constructively given. Moreover, a periodic consensus protocol is obtained as a special case and it is revealed that the maximum convergence speed can be achieved by choosing suitable gains. 9 Elsevier Inc. All rights reserved. 1. Introduction The consensus problem of networked multi-agent systems has attracted great attention of researchers in the control community, because of broad applications of cooperative control [1], formation control [ 4], flocking [5,6] and so on. The consensus problem has been considered for a long time, since Vicsek et al. [7] first proposed a discrete-time mode for multi-agent systems. Jadbabaie [8]gave a mathematical explanation for the consensus of the Vicsek model. Based on the algebraic graph theory, Olfati-Saber and Murray [9,1] discussed the consensus problem for networks of first order integrator agents. Directed networks with fixed/switched topologies and undirected networks with time-delays and fixed topologies were considered respectively. With simple linear consensus protocols, the agreement of agents were achieved under the assumption of strong connection of digraphs. Ren and Beard [11] generalized the results of [1] and presented more relaxed condition for the topology Corresponding author. address: zhujiandong@njnu.edu.cn (J. Zhu) /$ - see front matter 9 Elsevier Inc. All rights reserved. doi:1.116/j.laa

2 7 J. Zhu et al. / Linear Algebra and its Applications 431 (9) of directed networks, that is the existence of a spanning tree. Similar results were obtained by Lin et al. [1]. With the development of the issue, different models were considered and a lot of new consensus protocols appeared. Xiao and Wang [13] proposed a finite time consensus problem for networks of scalar integrators and designed a piecewise continuous nonlinear protocol. Profiri and Stilwell [14] considered the consensus problem for a group of agents that communicate via a stochastic information network and sufficient conditions for asymptotic almost sure consensus were given. Tian and Liu [15] considered the consensus of multi-agent systems with diverse input and communication delays and obtained sufficient conditions for the consensus through frequency-domain analysis. In the references on the consensus problem, multi-agent systems with double-integrator dynamics have been paid great attentions because of their importance in practice. Some special second-order consensus protocols were proposed and some consensus conditions were obtained [16 18]. In particular, Ren and Atkins [17] obtained two kinds of second-order consensus protocols, under which the state of each agent converges to a constant or a linear function with respect to the time. Zhang and Tian [] introduced the concept of consentability under the two kinds of second-order consensus protocols. For discrete multi-agent systems with double-integrator dynamics, Zhang and Tian obtained necessary and sufficient conditions of the consentability for fixed and stochastic switching topologies respectively. Recently, Ren [1] studied the synchronization of coupled second-order linear harmomic oscillators with local interaction, that means the state of each agent converges to the same periodic function. However, the protocols discussed in the existing literature has not address the consensus of positive exponential dynamics, namely, each agent s state converges to a positive exponential functions. In this paper, we will consider a more general form of consensus protocols for the multi-agent system with double-integrator dynamics. The advantage of the general consensus protocol is that different consensus dynamics including linear, periodic and positive exponential dynamics can be realized by choosing different consensus gains. As a special case, we obtain a simple periodic consensus protocol which is similar to the result for the fixed topology of [1], with which the only difference is that there is a gain used to adjust the convergence speed in our periodic consensus protocol. A very important problem in the study of the second-order consensus protocols is characterization of all the possible consensus gains, namely, finding necessary and sufficient conditions for any gain being a consensus gain. For the special consensus protocols proposed in [17], Ren and Atkins [17] only obtained sufficient conditions. For one consensus protocol in [17], Lin et al. [18] obtained a necessary and sufficient condition of the consensus. By converting the consensus problem for the original system into a stability problem for reduced order system, Zhang and Tian [] presented the necessary and sufficient condition of the consensus in the form of linear matrix inequality (LMI) for the two kinds of second-order consensus protocols in [17]. However, for the general protocol, the problem has not been solved. Another important problem is the transit performance of multi-agent systems. The performance of the first order consensus protocols was discussed by Olfati-Saber and Murray [1] and it is revealed that if the graph is undirected, then the convergence speed is determined by the algebraic connectivity, i.e. the second smallest eigenvalue of the Laplacian matrix. But for the second order consensus protocols, we have not find any reported results about the performance. In this paper, for the general protocol of the multi-agent systems with double-integrator dynamics, we obtain a necessary and sufficient condition for solving the consensus problem, which includes the results in [17,18] as special cases. All the possible consensus gains are described by two inequalities. Moreover, the convergence speed of systems are discussed. For the proposed periodic consensus protocol with an undirected graph, given the desired vibration frequency, the maximum convergence speed is achieved by choosing suitable gains, which is determined by the largest and the second smallest eigenvalues of the Laplacian matrix. Some illustrative examples show the effectiveness of the design method.. Problem statement Consider the networked multi-agent system with the fixed topology G = (V, E, A), which is a weighted digraph of order n with the set of nodes V ={v 1,v,...,v n }, set of edges E V V, and a

3 J. Zhu et al. / Linear Algebra and its Applications 431 (9) nonnegative adjacency matrix A = (a ij ).AnedgeofG is denoted by e ij = (v i,v j ), which means node v j can receive information from v i. Adjacency matrix A is defined such that a ij > ife ji E, while a ij = if e ji / E. We denote the set of neighbors of node v i by N i and assume N i ={v i1,v i,...,v ini }. The Laplacian matrix of the weighted digraph is defined as L = (l ij ), where l ii = n j=1,j /=i a ij and l ij = a ij (i /= j).let1 n denote the n 1 column vector of all ones. Let I n denote the n n identity matrix. It is obvious that L1 n =. Suppose each node of the graph is a dynamic agent with double-integrator dynamics ξ i = ζ i, (1) ζ i = u i, () where ξ i R, ζ i R, u i R and i = 1,,..., n. We consider the general form of the second-order consensus protocol u i = βξ i αζ i + v j N i a ij [γ (ξ j ξ i ) + γ 1 (ζ j ζ i )]. (3) Let ξ =[, ξ,..., ξ n ] T and ζ =[ζ 1, ζ,..., ζ n ] T. With the control protocol (3), the closed-loop systems of Eqs. (1) and () can be written in the matrix form as ] [ ] ξ = Γ, (4) [ ξ ζ ζ where [ ] n n I Γ = n βi n γ L αi n γ. (5) 1 L We say the consensus problem is solved if ξ i ξ j and ζ i ζ j ast for all i, j = 1,,..., n. The main goal of this paper is to find a necessary and sufficient condition of the consensus. 3. Necessary and sufficient conditions of the consensus To analyze the consensus of (4), we first research the eigenvalues of Γ. Denote the eigenvalues of L by μ 1 =, μ,..., μ n. With simple computation, we have det(λi n Γ ) = det[λ I n + (αi n + γ 1 L)λ + (βi n + γ L)] n = [λ + (α γ 1 μ i )λ + (β γ μ i )]. (6) i=1 Thus for each μ i, there exist two eigenvalues of Γ, denoted by λ i1 and λ i respectively. Since μ 1 =, we obtain two eigenvalues λ 11 and λ 1 of Γ : λ 11, 1 = α ± α 4β. (7) Zhang and Tian [] have obtained a necessary and sufficient condition for the consensus of discretetime multi-agent systems (see Theorem 1 of []). Here, with the same procedure, we give the continuous-time form of Theorem 1 of []: Lemma 1. The consensus is achieved if and only if Re(λ ij )<, i =, 3,..., n; j = 1,. (8) Proof. Let ξ =[ξ,..., ξ n ] T, ζ =[ζ,..., ζ n ] T,

4 74 J. Zhu et al. / Linear Algebra and its Applications 431 (9) x(t) = ξ 1 n 1, y(t) = ζ ζ 1 1 n 1. Then the consensus problem is equivalent to x(t) and y(t). Set [ ] [ ] 1 l11 ϕ P =, L =. ψ L Then and 1 n 1 I n 1 [ ] x ζ 1 = P ξ P][ ζ y PLP 1 = [ ] φ L (9) (1) by L1 n =, where L = L 1 n 1 ϕ. With the linear transformation (9), the closed-loop system (4) is transformed to 1 ζ 1 ẋ = β α γ 1 φ γ φ ζ 1 I n 1 x. ẏ β γ 1 L α γ L y Hence, the consensus is achieved if and only if (8) holds. Remark 1. The proof idea comes from [], that is, transforming the consensus problem to the stability of a reduced system. Moreover, using the method in the proof of Theorem 1 of [], we also can prove Lemma 1 and obtain the synchronizing state: x(t) = ( x 1 () p T e At)., x n () where p is the left eigenvector of L associated with eigenvalue satisfying p T 1 =, x i =[ξ i, ζ i ] T and [ ] 1 A = β α. As a matter of fact, x(t) is the solution of differential equation ẋ = Ax, x() =[p T ξ(), p T ζ()] T, which we call the group decision dynamics. In order to research the eigenvalues of the collective dynamics (4), by (6) we need to analyze the stability of a class of quadratic polynomials with complex coefficients. Lemma. Denote Re(μ) and Im(μ) by p and q respectively. The two roots of the polynomial f μ (λ) = λ + (α γ 1 μ)λ + β γ μ (11) lie in the open left-half complex plane if and only if α>pγ 1, (1) γ β> q (α pγ 1 ) γ γ 1 q + pγ. (13) α pγ 1

5 J. Zhu et al. / Linear Algebra and its Applications 431 (9) Proof. First of all, we can see that every root of f (λ) lies in the open left-half complex plane if and only if every root of the real coefficient polynomial f μ (λ)f μ (λ) does, since f μ (λ ) = f μ (λ ) for any complex number λ.leta = α pγ 1, b = β pγ, c = qγ 1 and d = qγ. With simple calculations, we have where f μ (λ)f μ (λ) = λ 4 + a 1 λ 3 + a λ + a 3 λ + a 4, (14) a 1 = a, a = a + b + c, (15) a 3 = ab + cd, a 4 = b + d. (16) By Hurwitz stability criteria, all the roots have negative real parts if and only if Δ 1 = a >, (17) Δ = a 1 1 a 3 a = (a 3 + ab + ac cd) >, (18) a 1 1 Δ 3 = a 3 a a 1 = a 4 a 3 4(a + c )[a b + (ac d)d] >, (19) Δ 4 = (b + d )Δ 3 >. Obviously, Eqs. (17) () are equivalent to a >, (1) b > cd a a c, () b > d a a, (3) b + d /=. (4) In the following, we will show that () and (4) can be removed. With simple computation, one has ( d cd ) ( ) cd a a a a c = a (d ac) + > a, (5) which means that () is implied by (3). Moreover, if b + d =, then b = d =, which is contradict to (3). Therefore, each root of (11) has a negative real part if and only if both (1) and (3), or equivalently (1) and (13), hold. Lemma 3. Assume p = Re(μ) <, q = Im(μ), γ and α. Then (1) and (13) hold if and only if γ > βp p + q, (6) γ 1 > α + γ q (α q + α q + 4β p) p β, p (7) where β = (β pγ )p γ q = βp γ (p + q ). Proof. If q = orγ =, the lemma is obvious. We assume q /= and γ /= in the following. Since p <, inequality (1) can be rewritten as γ 1 >α/p. Moreover, if (1) holds, then (13) can be rewritten as namely, (β pγ )(α pγ 1 ) + γ γ 1 q (α pγ 1 ) γ q >, (8) ()

6 76 J. Zhu et al. / Linear Algebra and its Applications 431 (9) β pγ 1 (β + γ q )αγ 1 + (β pγ )α γ q >. (9) (Necessity) Let γ q h(γ ) = (α pγ) γ γ q α pγ + pγ. (3) With simple calculations, we obtain dh dγ = γ q (α pγ)[α(α pγ) pγ ] < (α pγ) 4 (31) for all γ ( α p, + ), which implies h(γ ) is a decreasing function. By (13)wehave β>h(γ 1 )> lim h(γ ) = pγ + γ q. γ + p (3) Thus (6) holds and β <. Denote the left side of (9) byg(γ 1 ), which is a quadratic polynomial with respect to γ 1 with coefficient β p > and discriminant Δ = [ (β + γ q )α ] 4β p [ (β pγ )α γ q] = γ q (q α + 4β p)>. (33) Solving (9) yields γ 1 >ρ 1 or γ 1 <ρ, where ( ) ρ 1, = α γ q α q ± q α + 4β p + p pβ (34) are the two roots of polynomial g(γ 1 ).From(34), we can see ρ <α/p <ρ 1. Thus by γ 1 >α/p, we obtain γ 1 >ρ 1, which is (7). (Sufficiency) It is easy to see that (7) implies γ 1 >α/p, that is (1) holds. From (6), we have β <, which implies β p >. Thus (9) is obtained from γ 1 >ρ 1. Hence (13) holds. Remark. In Lemma 4.4 of [17], for the special case of α, β =, γ 1 > and γ = 1, only a sufficient condition was given, which can be rewritten as the following simple form: γ 1 > Re(μ). (35) Here, for the case of α, β = and γ 1 >, we obtain a necessary and sufficient condition from Lemma 3: γ 1 > α + αq + q q α 4p(p + q )γ. p p(p + q (36) ) In particular, if q =, then (36) holds for any γ 1 >, which implies Lemma 4. of [17]. Let α = β =, γ 1 > and γ >. From (36), we obtain the necessary and sufficient condition γ Im(μ) γ 1 > μ Re(μ), (37) which also includes the sufficient condition obtained by Ren [17] as a special case. Let α = β =, p < and γ 1 = 1. Then, by Lemma, polynomial (11) is stable if and only if <γ < (p + q )p q, (38) which is just Lemma of [18].

7 J. Zhu et al. / Linear Algebra and its Applications 431 (9) By Theorem 1 and Lemma, we obtain the following theorem: Theorem 1. Let p i = Re(μ i ) and q i = Im(μ i ). Then the consensus is achieved by the protocol (3) ifand only if α> max (p i γ 1 ), (39) i n ( γ β> max q i i n (α p i γ 1 ) γ ) γ 1 qi + p i γ. (4) α p i γ 1 Remark 3. Inequalities (39) and (4) describe all possible consensus gains. If α>, β> and conditions (39) and (4) hold, then all the eigenvalues of Γ have negative real parts, which implies the asymptotical stability. In this case, we call the consensus trivial consensus. Otherwise, we call non-trivial consensus. Remark 4. If α orβ, then the non-trivial consensus is achieved if and only if conditions (39) and (4) hold. One may wonder why it is not assumed that the direct graph G has a spanning tree as stated in [11]. Indeed, this condition is implied by (39) and (4) asα orβ. Assume G has no spanning tree, then by Lemma.3 of [11], there is a μ i = (i ), i.e. p i = q i =. Thus (39) and (4) imply α> and β>, which is a contradiction. Remark 5. If G has a spanning tree, then max i n p i <. Thus (39) is equivalent to α γ 1 > (41) max i n p i as γ 1 >. Therefore, for any given α and γ, there always exist sufficiently large numbers γ 1 and β such that (41) and (4) hold, that is, the consensus is achieved. Moreover, we can choose β such that 4β α >. Thus by (7), we have Re λ 11 = Re λ 1 = α/. That is if we hope the consensus poles are a pair of conjugated complex numbers, then their real part can be arbitrarily assigned. If we choose α<, then the positive exponential consensus is achieved. If we let α =, then the periodic consensus is achieved. In particular, we let α = γ =, then (41) and (4) are reduced to γ 1 > and β>. For this special case, we have following corollary, which is actually just Theorem 3.1 of [1]as γ 1 = 1. Corollary 1. The periodic consensus is achieved by the protocol u i = βξ i + a ij γ 1 (ζ j ζ i ) (4) v j N i if and only if γ 1 >, β> and G has a spanning tree. Moreover, the vibration frequency of the periodic consensus dynamics is ω = β and the error dynamics poles are determined by γ 1. Proof. Substituting α = γ = into (3) and (7), we obtain the consensus protocol (4) and the consensus poles λ 11,1 =± βi. From Remark 4 and Remark 5, we know that the periodic consensus is achieved if and only if γ 1 >, β>and G has a spanning tree. Thus the vibration frequency of the periodic consensus dynamics is ω = β.from(6), we know λ i1 and λ i are the roots of λ γ 1 μ i λ + β (i =,..., n). Thus, as the vibration frequency is given, the positions of the other eigenvalues except λ 11 and λ 1 are determined by γ 1. Therefore, the error dynamics poles are determined by γ 1. Theorem. Let p i = Re(μ i ) and q i = Im(μ i ). Assume γ, α and graph G has a spanning tree. Then the consensus is achieved by the protocol (3) if and only if γ > max i n βp i p i + q, (43) i

8 78 J. Zhu et al. / Linear Algebra and its Applications 431 (9) γ 1 > max i n where β ) i = βp i γ (p i + q i. α + γ αq i + γ q i q i α + 4β i p i p i β, (44) i p i Proof. Since G has a spanning tree, by Lemma.3 of [11], we have p i < for all i =, 3,..., n. Hence, by Theorem 1, Lemma and Lemma 3, the theorem is proved. Corollary. Assume γ >, α, β = and graph G has a spanning tree. Then the consensus is achieved by the protocol (3) if and only if γ 1 > max α + αq i + q i q i α 4p i γ (p i + q ) i i n p i p i (p i + q ). (45) i Specially, for α = β =, the consensus is achieved by the protocol (3) if and only if γ Im(μ i ) γ 1 > max i n μ i Re(μ i ). (46) Remark 6. Corollary 1 is also a corollary of Theorem. Corollary includes Theorem 4. and 4.3 of [17] as special cases. For the case of α = β =, Corollary is equivalent to Theorem 1 of [18]. Remark 7. From (7), we know α and β determine the poles of the consensus dynamics. If β<, then one consensus pole is positive. In this case, the positive exponential consensus is also achieved. As γ, α and β are given, γ 1 determines the poles of the error dynamics poles and the convergence speed. A natural question is how to choose γ 1 in ( ˆγ 1, + ) such that the maximum convergence speed of the error dynamics is achieved, where ˆγ 1 is the right side of (44). We solve the problem in the following. Moreover, ifγ can be changed, one interesting question is how the change ofγ affects the convergence speed. And it is more interesting to show how both γ and γ 1 affect the convergence speed. We will research the problems in the future work. In order to achieve the maximum convergence speed, we need to research the root locus of the complex coefficient polynomial λ + (α γ 1 μ)λ + (β γ μ) (47) as γ 1 varies from ˆγ 1 to +. Letz = A(cos θ + i sin θ) be any complex number, where A > and π <θ π. Denote z by ( θ z = A cos + θ ) i sin. (48) ( ) Since π/ <θ/ π/, we have Re z. Hence, the two roots of (47) can be expressed as λ 1, (γ 1 ) = (α γ 1μ) ± (α γ 1 μ) 4(β γ μ). (49) Lemma 4. Assume p = Re(μ) <, q = Im(μ), γ, α, (6) and (7) hold. Denote the right side of (7) by γ 1. Then and lim γ 1 γ 1 + Reλ 1(γ 1 ) =, lim γ 1 + arg λ (γ 1 ) = arg μ, lim λ 1(γ 1 ) =, (5) γ 1 + lim λ (γ 1 ) =+. (51) γ 1 +

9 J. Zhu et al. / Linear Algebra and its Applications 431 (9) Im 4 Re Fig. 1. The root locus of λ + (α γ 1 μ)λ + (β γ μ). 4 Proof. From (49), we have Re(λ ) Re(λ 1 ). By Lemma 3 and the continuity of the roots of a polynomial with respect to its coefficients, we obtain Reλ 1 (γ 1 ) (γ 1 γ 1 +). By(49), it is obtained that (β γ μ) λ 1 (γ 1 ) = (α γ 1 μ) + (α γ 1 μ) 4(β γ μ). (5) Denote the denominator of the right side of (5) byφ(γ 1 ). Then φ(γ 1 ) Reφ(γ 1 ) Re(α γ 1 μ) = α γ 1 p +, (53) as γ 1 +. Thus λ 1 (γ 1 ) (γ 1 + ). Since λ 1 (γ 1 ) + λ (γ 1 ) = (α γ 1 μ), wehave lim [λ λ (γ 1 ) (γ 1 ) (γ 1 μ α)] =, lim = μ. (54) γ 1 + γ 1 + γ 1 From (54), we obtain (51). Remark 8. For α =, β = 36, γ = 1 and μ =. +.1i, we plot the root locus of (47) as shown in Fig. 1, which validates Lemma 4. Theorem 3. Assume all the conditions of Theorem are satisfied. Let c(γ 1 ) = max{reλ ij i =, 3,..., n; j = 1,.} and ˆγ 1 be the right side of (44). Then there must exists γ 1 ( ˆγ 1, + ) such that c(γ 1 ) = min γ 1 ( ˆγ 1,+ ) c(γ 1), (55) namely, there exists γ 1 ( ˆγ 1, + ) such that the maximum convergence speed is achieved as γ 1 = γ 1 for given α, β and γ. Proof. By Theorem and Theorem 1, we have c(γ 1 )<. From Lemma 4, it is obtained that there is at least one λ i1 (t) satisfying lim γ 1 ˆγ 1 + Reλ i1(γ 1 ) =, which imply lim λ i1(γ 1 ) =, (56) γ 1 + lim c(γ 1) = lim c(γ 1) =. (57) γ 1 ˆγ 1 + γ 1 +

10 71 J. Zhu et al. / Linear Algebra and its Applications 431 (9) Fig.. Root locus of λ γ 1 μ i λ + β. Fig. 3. γ 1 is obtained as Reλ 1 = λ n1. By the continuity of c(γ 1 ) with respect to γ 1, the theorem is proved. Remark 9. Theorem 3 reveals the existence of γ 1. Of course, we can get an approximate value of γ 1 by an optimization algorithm. A natural question is whether we can get an analytical expression of γ 1. In the following theorem, we consider the problem for a special case. Theorem 4. Assume G is a connected undirected graph with eigenvalues = μ 1 >μ μ 3 μ n. Under the consensus protocol (4), as the vibration frequency ω = β is given, the maximum convergence speed of the error dynamics is achieved as γ 1 = β μ (μ μ n ). (58) Proof. Firstly, it is easy to see all the polynomials λ γ 1 μ i λ + β(i =, 3,..., n) have the same root locus shown in Fig.. Secondly, we show that if >μ k >μ l, then λ l1 (γ 1 ) runs faster than λ k1 (γ 1 ) along the root locus as γ 1 varies from to. Indeed, as γ 1 β β μ k μ l,bothλ k1 and λ l1 are real numbers and β λ k1 = γ 1 ( μ k ) + γ1 μ 4β < β k γ 1 ( μ l ) + γ 1 μl 4β = λ l1. (59) As γ 1 β β μ l μ k,bothλ k1 and λ l1 are imaginary numbers and

11 J. Zhu et al. / Linear Algebra and its Applications 431 (9) Reλ k1 = γ 1μ k > γ 1μ l Fig. 4. The directed graph of the multi-agent system. = Reλ l1. (6) Hence, in set {λ i1 i =, 3,..., n}, the eigenvalue λ n1 is the fastest and λ 1 the slowest along the same root locus. Finally, by the root locus, we can see c(γ 1 ) = max{reλ 1, Reλ n1 }. Then the minimum of c(γ 1 ) is achieved as Reλ 1 = λ n1 (see Fig. 3), that is, γ 1 μ = γ 1μ n + γ1 μ 4β n. (61) Solving (61), we obtain (58). Remark 1. Theorem 4 reveals that as the desired vibration frequency is given, the maximum convergence speed is determined by the largest and the second smallest eigenvalues of the Laplacian matrix. 4. Simulations Consider the multi-agent systems with double-integrator dynamics (1) and () and graph shown in Fig. 4. Obviously, the Laplacian matrix of G is 1 ζ 1,..., ζ t t Fig. 5. The periodic consensus with desired period.

12 71 J. Zhu et al. / Linear Algebra and its Applications 431 (9) γ 1 = γ 1 = γ =5 1 Fig. 6. The maximum convergence speed is achieved as γ 1 = t 8 6 ζ 1,..., ζ t Fig. 7. The positive exponential consensus. L = (6) We first consider the periodic consensus protocol (4). Let the desired period of consensus dynamics be T =. Then β = (π/t).letγ 1 = 1 and the initial value vector be [ξ T () ζ T ()] =[ ]. Fig. 5 shows the periodic consensus of ξ i and ζ i.

13 J. Zhu et al. / Linear Algebra and its Applications 431 (9) γ = γ 1 = γ 1 =.6 Fig. 8. From disagreement to agreement of ξ i. ζ 1,..., ζ γ 1 =.3 ζ 1,..., ζ γ 1 =.469 ζ 1,..., ζ γ =.6 1 Fig. 9. From disagreement to agreement of ζ i. If we regard the graph shown in Fig. 4 as an undirected graph, then the Laplacian matrix of G is L = (63) 1 3 With simple calculations, we obtain μ =.7639 μ 5 = Hence by (58), the maximum convergence speed is achieved as γ 1 = The simulation can be seen in Fig. 6 for different values of γ 1 with the same initial value vector [ξ T () ζ T ()] =[ ].

14 714 J. Zhu et al. / Linear Algebra and its Applications 431 (9) For the directed graph shown in Fig. 4, assume the desired poles of the consensus dynamics are.1 and. Then α = 1.9 >, β =.. With simple calculation, the right side of (43) isequalto.571. We let γ = 1 >.571. Then the right side of (44) equals.17. Hence, by Theorem, the positive exponential consensus is achieved for any γ 1 >.17. Fig. 7 shows the simulation for γ 1 =.1 and the initial value vector [ ]. Let α = β = and γ 1 = 1. Then the right side of (46) equals.469. Hence, by Corollary, the consensus is achieved if and only if γ 1 >.469. Using (11) in Theorem 4. of [17], one obtains γ 1 > 1, which is only a sufficient condition. Figs. 8 and 9 show the phenomena from disagreement to agreement. 5. Conclusions For the general protocol of the multi-agent systems with double-integrator dynamics, we obtain a necessary and sufficient condition for solving the consensus problem, which implies all the possible consensus gains can be described. With the general protocol, different forms of consensus dynamics including linear, periodic and positive exponential dynamics can be realized. For the proposed periodic consensus protocol with an undirected graph, given the desired vibration frequency, the maximum convergence speed is achieved by choosing suitable gains, which is determined by the largest and the second smallest eigenvalues of the Laplacian matrix. Some illustrative examples show the effectiveness of the design method. Acknowledgements The authors would like to thank the National Natural Science Foundation of China (under the Grant 1714, 64538), National 863 Programme of China (under Grant 6AA4Z63), Tianyuan Foundation for Mathematics (under the Grant 1565) for financial supports for this work. References [1] P.K.C. Wang, F.Y. Hadaegh, Coordination and control of multiple microspacecraft moving in formation, J. Astronaut. Sci., 44 (3) (1994) [] T. Balch, R.C. Arkin, Behavior-based formation control for multirobot terms, IEEE Trans. Robot. Automat. 14 (6) (1998) [3] X. Hu, Formation control with virtual leaders and reduced communications, IEEE Trans. Robot. Automat. 17 (6) (1) [4] A. Fax, R.M. Murray, Information flow and cooperative control of vehicle formations, IEEE Trans. Automat. Contr. 49 (9) (4) [5] J. Toner, Y. Tu, Flocks, herds, and schools: a quantitative theory of flocking, Phys. Rev. W 58 (4) (1998) [6] R. Olfati-Saber, R.M. Murray, Flocking for multi-agent dynamic systems: algorithms and theory, IEEE Trans. Automat. Contr. 51 (3) (6) [7] T. Vicsek, A. Czirok, E. Ben-Jacob, et al., Novel type of phase transitions in a system of self-driven particles, Phys. Rev. Lett. 75 (6) (1995) [8] A. Jadbabaie, J. Lin, A.S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans. Automat. Contr. 48 (6) (3) [9] R.O. Saber, R.M. Murray, Consensus protocols for networks of dynamic agents, in: Proceedings of Am. Contr. Conf., 3, pp [1] R. Olfati-Saber, R.M. Murray, Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans. Automat. Contr. 49 (9) (4) [11] W. Ren, R.W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Trans. Automat. Contr. 5 (5) (5) [1] Z.Y. Lin, B. Francis, M. Maggiore, Necessary and sufficient graphical condition for formation control of unicycles, IEEE Trans. Automat. Contr. 5 (1) (5) [13] F. Xiao, L. Wang, Reaching agreement in finite time via continuous local state feedback, in: Proc. the 6th Chinese Contr. Conf., Zhangjiajie, Hunan, PR China, 7, pp [14] M. Profiri, D.J. Stilwell, Consensus seeking over random weighted directed graphs, IEEE Trans. Automat. Contr. 5 (9) (7) [15] Y.-P. Tian, C.-L. Liu, Consensus of multi-agent systems with diverse input and communication delays, IEEE Trans. Automat. Contr. 53 (9) (8) [16] G. Xie, L. Wang, Consensus control for a class of networks of dynamic agents, Int. J. Robust Nonlinear Contr. 17 (1 11) (7)

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