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Commun. Theor. Phys. (Beijing, China) 51 (2009) pp. 101–109 c Chinese Physical Society and IOP Publishing Ltd

Vol. 51, No. 1, January 15, 2009

Second-Order Consensus of Multiple Agents with Coupling Delay SU Hou-Sheng∗ and ZHANG Wei Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China

(Received April 11, 2008)

Abstract In this paper, we investigate two kinds of second-order consensus algorithms for multiple agents with coupling delay under general fixed directed information topology. Stability analysis is performed based on Lyapunov– Krasovskii functional method. Delay-dependent asymptotical stability condition in terms of linear matrix inequalities (LMIs) is derived for the second-order consensus algorithm of delayed dynamical networks. Both delay-independent and delay-dependent asymptotical stability conditions in terms of LMIs are derived for the second-order consensus algorithm with information feedback. PACS numbers: 05.45.-a

Key words: consensus, time delay, linear matrix inequality, multi-agent systems

1 Introduction Recently, consensus problems have attracted much attention among researchers studying biology, physics, computer science, and control engineering.[1] This is partly due to broad applications of multi-agent systems in many areas including cooperative control of mobile robots, unmanned air vehicles (UAVs), autonomous underwater vehicles (AUVs), automated highway systems, and so on. First-order consensus problems for networked dynamic systems have been extensively studied by many researchers.[2] In reality, however, a broad class of agents should be described by second-order dynamic models, such as torque motor and gas jet, which are adjusted for their desired motion directly by their acceleration rather than by their speeds. Hence, the consensus problem of the multi-agent system modeled by double integrators is more challenging. Ren et al. proposed a set of second-order consensus algorithms,[3] which include fundamental consensus algorithm and consensus algorithm with information feedback under directed networks. Lee and Spong proposed a second-order consensus algorithm of multiple inertial agents on balanced graphs.[4] Due to the finite speeds of transmission and spreading as well as traffic congestions, there usually are time delays in spreading and communication in reality. Therefore, time delay should be considered in designing the consensus algorithms. First-order consensus algorithms with time delays have been broadly investigated.[5−8] In Ref. [5], time-varying delays of first-order consensus are considered for directed networks. In Ref. [6], average consensus in directed networks of agents with coupling timedelay is investigated. In Ref. [7], uniformly constant delay is considered in directed networks. In Ref. [8], time delay problems have been studied for the discrete consensus algorithm. However, less attention has been paid to the delayrelated consensus problem for multiple agents with double ∗ Corresponding

author, E-mail: [email protected]

integrator dynamics. In this paper, we study the secondorder consensus algorithms with a coupling delay under general directed networks. Simple second-order consensus algorithms analogous to that in Refs. [3] and [4] are applied to solving the consensus problem of the multi-agent system with a coupling delay. By using the passivity decomposition methods in Ref. [4], we decompose the system into two subsystems: locked system dynamics and shape system dynamics. Based on the shape system dynamics and Lyapunov–Krasovskii functional method,[9−11] a convergence condition for delay-dependent asymptotical stability in terms of LMIs is derived for fundamental second-order consensus in time delayed networks. Moreover, based on Lyapunov–Krasovskii functional method, convergence conditions for both delay-independent and delay-dependent asymptotical stabilities in terms of LMIs are derived for second-order consensus algorithm with information feedback in time delayed networks. Numerical simulations are worked out to illustrate our theoretical results. An outline of this paper is as follows. In Sec. 2, we formulate the problem to be investigated. Consensus conditions for both delay-independent and delay-dependent asymptotical stabilities in terms of linear matrix inequalities are introduced in Sec. 3. Numerical examples are given in Sec. 4 to illustrate the theoretical results. Concluding remarks are stated in Sec. 5.

2 Problem Formulations The interaction topology of a network of agents is represented by a directed graph G = {V, E, W } with the set of nodes V = {n1 , . . . , nN }, edges E ⊆ V × V and W : E → R+ is a map assigning a positive weight to each edges. In this paper, we exclude the self-joining edges from E.

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The second-order agent dynamics is modeled by x˙ i = vi , n

v˙ i = ui ,

i = 1, . . . , N ,

n

(1)

n

where xi ∈ R , vi ∈ R , and ui ∈ R , and a second-order consensus protocol is proposed as follows,[3,4] N X ui = − gij wij [(xi − xj ) + γ(vi − vj )], i = 1, . . . , N, (2) j=1

where gij = 1 if information flows from agent i to agent j and 0 otherwise, ∀i 6= j, and weight wij > 0 and scaling factor γ > 0 are uniformly bounded. The adjacency matrix A of the information exchange topology is defined accordingly as aii = 0 and aij = gij wij , ∀i 6= j. The goal of consensus protocol (2) is to guarantee that kxi − xj k → 0 and kvi − vj k → 0 as t → ∞. In order to achieve a desired value, a second-order consensus protocol with information feedback is proposed as follows,[3] ui = v˙ d − β[(xi − xd ) + γ(vi − vd )] −

N X

gij wij [(xi − xj )+γ(vi −vj )] ,

i = 1, . . . , N , (3)

j=1

where β > 0 are uniformly bounded, xd ∈ Rn and vd ∈ Rn are desired value for xi and vi respectively and satisfy x˙ d = vd . The goal of consensus protocol (3) is to guarantee that x1 = · · · = xN = xd and v1 = · · · = vN = vd as t → ∞. Time delays commonly exist in the real world. In this paper, according to protocols (2) and (3), we consider second-order consensus algorithms with a coupling delay for weighted general directed networks, which can be written as N X ui = − gij wij [(xi (t − τ ) − xj (t − τ )) j=1

+ γ(vi (t − τ ) − vj (t − τ ))] ,

(4)

and ui = v˙ d − β[(xi − xd ) + γ(vi − vd )] −

N X

gij wij [(xi (t − τ ) − xj (t − τ ))

j=1

+ γ(vi (t − τ ) − vj (t − τ ))] ,

(5)

Equations (4) and (5) can be written in matrix form as u = v˙ = −(L ⊗ In )x(t − τ ) − γ(L ⊗ In )v(t − τ ) , and u = v˙ d ⊗ 1N − β(x − xd ⊗ 1N ) − γβ(v − vd ⊗ 1N ) − (L ⊗ In )x(t − τ ) − γ(L ⊗ In )v(t − τ ) , where x = [x1 , . . . , xN ]T , v = [v1 , . . . , vN ]T , L = [lij ] is P given as lii = j6=i gij wij and lij = −gij wij , ∀i 6= j, and τ is the time delay (we assume that all the delays are the same in the network). Hereafter, consensus protocol (4) (or consensus protocol (5)) is said to achieve asymptotic consensus if kxi − xj k → 0 and kvi − vj k → 0 as t → ∞

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(or if x1 = · · · = xN = xd and v1 = · · · = vN = vd as t → ∞). Before stating the main results of this paper, we need the following preliminaries: Lemma 1[10] Suppose that a symmetric matrix is partitioned as   E1 E2 E= , E2T E3

where E1 and E3 are square. E is positive definite if and only if both E1 and E3 − E2T E1−1 E2 are positive definite. Lemma 2[11] Suppose that a and b are vectors, then for any positive-definite matrix E, the following inequality holds: −2aT b ≤ inf {aT Ea + bT E −1 b} . E>0

3 Main Results Using the Lyapunov–Krasovskii functional approach, some sufficient conditions for ensuring the stability of the consensus states of protocols (4) and (5) are derived in this section. 3.1 Fundamental Second-Order Consensus of Delayed Dynamical Networks Following the passive decomposition,[4] we define the following coordinate transformation:  1 1 1 1  ... N N N N    1 −1 0 . . . 0    z = (S ⊗ In )x; S =  1 −1 . . . 0   0  , (6)   · .. .. .. ·   · . . . · · · 0 . . . . . . 1 −1 T where z = [z1 , z2 , . . . , zn ] ∈ RnN is the transformed coordinate, and S ∈ RN ×N is the (full-rank) transformation matrix. Let us define ze = [z2 , z3 , . . . , zN ]T ∈ Rn(N −1) so that z = [z1 , zeT ]T . Then, from Eq. (6), we can show that ze describes the internal group shape as it is given by ze = [x1 − x2 , x2 − x3 , . . . , xn−1 − xn ]T , (7) and z1 abstracts the overall group maneuver, as N 1 X z1 = xi . N i=1

(8)

Using Eq. (6), we can rewrite the protocol (4) with respect to z such that ((S −T S −1 ) ⊗ In )¨ z + γ((S −T LS −1 ) ⊗ In )z(t ˙ − τ)

+ ((S −T LS −1 ) ⊗ In )z(t − τ ) = 0 . (9) And we can get that  ¯T  0 D −T −1 −T −1 ¯ S S = diag [N, S] , S LS = ¯ , 0(N −1)×1 L where S¯ ∈ R(N −1)×(N −1) is a symmetric and positive¯ ∈ RN −1 is given definite matrix, the j-th component of D by N X ¯j = − D (L1k + L2k + · · · + Lnk ) , k=j+1

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Second-Order Consensus of Multiple Agents with Coupling Delay

j ∈ {1, 2, . . . , N − 1}, and the ij-th component of the ma¯ ∈ R(N −1)×(N −1) is given by trix L N N   X X ¯ ij = N − i D ¯j + L Lpq , N p=i+1 q=j+1

103

(i) For arbitrary initial conditions and scaling factor γ, the weighted average value x ¯˙ (t) is invariant, i.e. x ¯˙ (t) =

N X

ξi x˙ i (t) = x ¯˙ (0) ,

∀t ≥ 0 .

(12)

i=1

i, j ∈ {1, 2, . . . , N − 1}, and protocol (4) can be decomposed as the locked system dynamics ¯ T ⊗In )z˙e (t−τ )+(D ¯ T ⊗In )ze (t−τ ) = 0 , (10) N z¨1 (t)+γ(D and the shape system dynamics ¯ ⊗ In )z˙e (t − τ ) (S¯ ⊗ In )¨ ze (t) + γ(L ¯ ⊗ In )ze (t − τ ) = 0 . + (L (11)

(ii) If the following linear time-varying delayed differential equation is asymptotically stable about its zero solution:

Denote the weighted average value of the differential vector x˙ as N X x ¯˙ (t) = ξi x˙ i (t) ,

w(t) = [zeT , z˙eT ]T ,   0 0 B= ¯ ⊗ In ) −γ(S¯ ⊗ In )−1 (L ¯ ⊗ In ) −(S¯ ⊗ In )−1 (L   0 I(N −1)×(N −1) ⊗ In C= , 0 0

i=1

where ξ = [ξ1 , ξ2 , . . . , ξn ]T ∈ Rn is the left eigenvector of L Pncorresponding to eigenvalue of zero. And we assume i=1 ξi = 1. Theorem 1 Consider a system of N agents with dynamics (1), each steered by protocol (4).

w(t) ˙ = Cw(t) + Bw(t − τ ) ,

(13)

where

then the consensus states (kxi − xj k → 0 and kvi − vj k → 0 as t → ∞ for all i, j ∈ {1, 2, . . . , N }) are asymptotically stable.

Proof (i) From Eq. (4), we have (ξ ⊗ In )T u = (ξ ⊗ In )T x ¨ = −(ξ ⊗ In )T (L ⊗ In )x(t − τ ) − γ(ξ ⊗ In )T (L ⊗ In )x(t ˙ − τ) = 0 . Then, the weighted average value x ¯˙ (t) is invariant. (ii) From the shape system dynamics (11), we can get that ¯ ⊗ In )z˙e (t − τ ) + (S¯ ⊗ In )−1 (L ¯ ⊗ In )ze (t − τ ) , z¨e (t) = (S¯ ⊗ In )−1 γ(L (14) and then we can get Eq. (13). From Eq. (7), if the linear time-varying delayed differential equation (13) is asymptotically stable about its zero solution, then the protocol (4) is said to achieve (asymptotical) consensus. Based on the Lyapunov–Krasovskii functional method, a convergence condition for delay-dependent asymptotical stability in terms of linear matrix inequality is derived for consensus protocol (4). Theorem 2 Suppose that the time-invariant delay τ ∈ [0, h] for some h < ∞, if there exist some symmetric matrices P > 0, Q > 0, such that " # P (C + B) + (C + B)T P + 2h(C T QC + B T QB) P B < 0, (15) 1 BTP − Q h then the consensus states of consensus protocol (4) are asymptotically stable for any time-delay τ ∈ [0, h]. Proof Choose a Lyapunov–Krasovskii functional, V = V1 + V2 + V3 , where V1 = wT (t)P w(t) , Z τ Z t dβ w˙ T (α)Qw(α)dα ˙ , V2 = 0

V3 = 2

t−β

Z

t

τ wT (α)B T QBw(α)dα .

t−τ

(16)

The equation in system (13) can be written as Z t w(t) ˙ = (C + B)w(t) − B w(α)dα ˙ , t−τ

and thus, the derivate of V1 satisfies V˙ 1 = wT (t)((C + B)T P + P (C + B))w(t) Z t − [w˙ T (α)B T P w(t) + wT (t)P B w(α)]dα ˙ , t−τ

V˙ 2 = τ w˙ T (t)Qw(t) ˙ −

Z

t

w˙ T (α)Qw(α)dα ˙ .

t−τ

Furthermore

− w˙ T (α)B T P w(t) − wT (t)P B w(α) ˙ − w˙ T (α)Qw(α) ˙ T −1 = −[B T P w(t) + Qw(α)] ˙ Q [B T P w(t) + Qw(α)] ˙ + wT (t)P BQ−1 B T P w(t) .

But τ w˙ T (t)Qw(t) ˙ + V˙ 3 = τ [wT (t)C T QCw(t) + wT (t − τ )B T QBw(t − τ ) + wT (t)C T QBw(t − τ )

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Vol. 51

+ wT (t − τ )B T QCw(t)] + 2τ wT (t)B T QBw(t) − 2τ wT (t − τ )B T QBw(t − τ ) . Note that [wT (t)C T QCw(t) + wT (t − τ )B T QBw(t − τ ) + wT (t)C T QBw(t − τ ) + wT (t − τ )B T QCw(t)] − 2wT (t − τ )B T QBw(t − τ ) − 2wT (t)B T QBw(t) = −wT (t)C T QCw(t) − wT (t − τ )B T QBw(t − τ ) + wT (t)C T QBw(t − τ ) + wT (t − τ )B T QCw(t) = −(Cw(t) − Bw(t − τ ))T Q(Cw(t) − Bw(t − τ )) . As a result, we obtain V˙ = wT (t)((C + B)T P + P (C + B) + 2τ C T QC + 2τ B T QB + τ P BQ−1 B T P )w(t) −

Z

t T −1 [B T P w(t) + Qw(α)] ˙ Q

t−τ

T × [B T P w(t) + Qw(α)]d ˙ α − τ (Cw(t) − Bw(t − τ )) Q(Cw(t) − Bw(t − τ )) .

From the Shur complements (Lemma 1), this derivative is negative under condition (15). This completes the proof of Theorem 2. Remark 1 From Eq. (15), the convergence condition for fundamental second-order consensus of delayed dynami¯ and γ. Hence, the stacal networks is determined by L bilization of protocol (4) is determined by the topological structure, coupling weight and scaling factor. 3.2 Second-Order Consensus with Information Feedback of Delayed Dynamical Networks Denote the difference vectors between agent i and the reference as x ˜i = xi − xd and v˜i = vi − vd , then x ˜˙ i = v˜i , u ˜i = v˜˙ i = v˙ i − v˙ d , (18) and the protocol (5) can be written as u ˜ = − β[˜ x + γ˜ v ] − (L ⊗ In )˜ x(t − τ )

(17)

terms of linear matrix inequality are derived for consensus protocol (5). Theorem 3 If there exist two symmetric matrices P > 0, Q > 0, such that   P M + M TP + Q P N < 0, (22) N TP −Q then the consensus states of consensus protocol (5) are asymptotically stable for any time-delay τ > 0. Proof Choose a Lyapunov–Krasovskii functional Z t T V = J (t)P J(t) + J T (α)QJ(α)dα . (23) t−τ

Clearly, V is positive-definite. The derivative of V along the trajectories of function (21) is ˙ V˙ = J˙T (t)P J(t) + J T (t)P J(t)

+ J T (t)QJ(t) − J T (t − τ )QJ(t − τ ) − γ(L ⊗ In )˜ v (t − τ ) . (19) Equation (18) can be written in matrix form as = J T (t)[P M + M T P + Q]J(t) ˙    x ˜ 0 IN ⊗ In x ˜ + 2J T (t)P N J(t − τ ) − J T (t − τ )QJ(t − τ ) . ˙v˜ = −βIN ⊗ In −βγIN ⊗ In v˜ From Lemma 2, we have    0 0 x ˜(t − τ ) + , (20) 2J T (t)P N J(t − τ ) ≤ J T (t − τ )QJ(t − τ ) v˜(t − τ ) −L ⊗ In −γL ⊗ In + J T (t)P N Q−1 N T P J(t) , and equation (18) can be written as the form ˙ = M J(t) + N J(t − τ ) , J(t) (21) and then we can get where V˙ ≤ J T (t)[P M + M T P + Q + P N Q−1 N T P ]J(t) . T T T J(t) = [˜ x , v˜ ] , From Lemma 1, the LMI (22) is equivalent to   0 IN ⊗ In M= , P M + M T P + Q + P N Q−1 N T P < 0 . −βIN ⊗ In −βγIN ⊗ In   This completes the proof of Theorem 3. 0 0 N= . The delay-independent is very conservative if the delay −L ⊗ In −γL ⊗ In is already known and small. In the following, we will proClearly, if eqaution (21) is asymptotically stable about its zero solution, then the consensus state for protocol (5) is vide a delay-dependent condition for protocol (5), which is less conservative than the delay-independent one. asymptotically stable. Based on the Lyapunov–Krasovskii functional method, Theorem 4 Suppose that the time-invariant delay τ ∈ convergence conditions for delay-independent asymptoti- [0, h] for some h < ∞, if there exist some symmetric macal stability and delay-dependent asymptotical stability in trices P > 0, Q > 0, such that " # P (M + N ) + (M + N )T P + 2h(M T QM + N T QN ) P N < 0, (24) 1 N TP − Q h

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then the consensus states of consensus protocol (5) are asymptotically stable for any time-delay τ ∈ [0, h]. Proof Choose a Lyapunov–Krasovskii functional V = V1 + V2 + V3 , (25) where V1 = J T (t)P J(t) , Z τ Z t ˙ V2 = dβ J˙T (α)QJ(α)dα , 0

V3 = 2

t−β

Z

t

τ J T (α)N T QN J(α)dα .

t−τ

The following proof is similar to that of Theorem 2, and is omitted here. Remark 2 From Eqs. (22) and (23), the convergence conditions for second-order consensus with information feedback of delayed dynamical networks are determined by L, β, and γ. Hence, the stabilization of protocol (5) is determined by the topological structure, coupling weight and scaling factor.

4 Numerical Examples The simulations are performed with four agents in the plane. Initial values of xi and vi denote the position and velocity vectors of the agents, which are randomly chosen from box [0, 50]×[0, 50] and [0, 4]×[0, 4], respectively. Assume that the information graph is a cyclic graph, and the adjacency matrix A of the information exchange topology is   0 0 0 1 1 0 0 0   A= . 0 1 0 0 0 0 1 0

105

4.1 Fundamental Second-Order Consensus of Delayed Dynamical Networks Based on the fundamental second-order consensus protocol with a coupling delay, when the weight strength wij = 2, the scaling factor γ = 1, using Theorem 2, we can get a bound for the time delay as τ = 0.096, guaranteeing the asymptotical stability of the consensus states of the protocol (4). For example, if τ = 0.093 (< 0.096), by using the MATLAB LMI Toolbox, we find that there exist two positive-definite matrices,   101.04 40.69 33.82 20.93 −1.59 1.55  40.69 98.52 29.6 13.79 23.27 −1.27    33.82 29.6 90.8 12.11 22.7 25.49    P = ⊗ I ,  20.92 13.79 12.11 35.03 19.74 15.77  2    −1.59 23.27 22.7 19.74 47.33 21.48  1.55

−1.27

25.49 15.77

21.48

41.12  5.47 10.48   17.16    ⊗ I2 , 9.92   11.04 

28.83 10.72 12.8 17.76 10.89  10.72 29.52 10.51 10.76 22.83   12.79 10.51 28.91 5.54 10.76  Q=  17.76 10.76 5.54 22.42 11.15   10.89 22.82 10.76 11.15 23.79 5.47 10.48 17.16 9.91 11.04 22.24 such that condition (15) holds. Figure 1 shows fundamental second-order consensus with a coupling delay on the cyclic graph for the case τ = 0.093. Figures 1(a) and 1(b) plot the curves of the values of xi , which are convergent for x-axis and y-axis, respectively. Figures 1(c) and 1(d) depict the curves of the values of vi , which are convergent for x-axis and y-axis, respectively. 

Fig. 1 Fundamental second-order consensus with a coupling delay on the cyclic graph (τ = 0.093).

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Figure 2 shows fundamental second-order consensus with a coupling delay on the cyclic graph for the case τ = 0.18 (> 0.096). Figures 2(a) and 2(b) plot the curves of the values of xi , which are divergent for x-axis and y-axis, respectively. Figures 2(c) and 2(d) depict the curves of the values of vi , which are divergent for x-axis and y-axis, respectively.

Fig. 2 Fundamental second-order consensus with a coupling delay on the cyclic graph (τ = 0.18).

4.2 Second-Order Consensus with Information Feedback of Delayed Dynamical Networks In the following simulations, initial values of reference xd and vd are selected as (1,1) and (3,3) , respectively, and v˙ d = (0.1, 0.1). Based on the second-order consensus protocol (5), when the weight strength wij = 2, the scaling factor γ = 1 and β = 5, by using the MATLAB LMI Toolbox, we find that there exist two positive-definite matrices for Theorem 3,   1.31 0.51 0.05 0.51 0.18 0.03 −0.02 0.03  0.51 1.31 0.51 0.05 0.03 0.18 0.03 −0.02     0.05 0.51 1.31 0.51 −0.02 0.03 0.18 0.03       0.51  0.05 0.51 1.31 0.03 −0.02 0.03 0.18   ⊗ I2 , P = 0.03 −0.02 0.03 0.24 0.03 −0.03 0.03   0.18     0.03 0.18 0.03 −0.02 0.03 0.24 0.03 −0.03     −0.02 0.03 0.18 0.03 −0.03 0.03 0.24 0.28  0.03 −0.02 0.03 0.18 0.03 −0.03 0.03 0.24   0.98 0.11 −0.11 0.11 0.41 −0.13 −0.15 −0.13  0.11 0.98 0.11 −0.11 −0.13 0.41 −0.13 −0.15     −0.11 0.11 0.98 0.11 −0.15 −0.13 0.41 −0.13        0.11 −0.11 0.11 0.98 −0.13 −0.15 −0.13 0.41  ⊗ I2 ,  Q= 0.12 −0.11 0.12    0.41 −0.13 −0.15 −0.13 0.95    −0.13 0.41 −0.13 −0.15 0.12 0.95 0.12 −0.11     −0.15 −0.13 0.41 −0.13 −0.11 0.12 0.95 0.12  −0.13 −0.15 −0.13 0.41 0.12 −0.11 0.12 0.95 such that condition (22) holds. Figure 3 shows second-order consensus with information feedback and a coupling delay on the cyclic graph for the case that β = 5 and τ = 1. Figures 3(a) and 3(b) plot the curves of the values of xi , which

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are convergent for x-axis and y-axis, respectively. Figures 3(c) and 3(d) depict the curves of the values of vi , which are convergent for x-axis and y-axis, respectively.

Fig. 3 Second-order consensus with information feedback and a coupling delay on the cyclic graph (τ = 1, β = 5).

Figure 4 shows second-order consensus with information feedback and a coupling delay on the cyclic graph for the case that β = 2 and τ = 1. Figures 4(a) and 4(b) plot the curves of the values of xi , which are divergent for x-axis and y-axis, respectively. Figures 4(c) and 4(d) depict the curves of the values of vi , which are divergent for x-axis and y-axis, respectively.

Fig. 4 Second-order consensus with information feedback and a coupling delay on the cyclic graph (τ = 1, β = 2).

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For the case that β = 2, Theorem 3 fails to verify that the consensus states of protocol (5) are asymptotically stable. However, using Theorem 4, we can get a bound for the time delay as τ = 0.225, guaranteeing the asymptotical stability of the consensus states of the protocol (5). For example, for τ = 0.2, by using the MATLAB LMI Toolbox, we get the following positive-definite matrices for Theorem 4,   25.2 8.99 4.28 8.99 6.72 1.44 0.79 2.48  8.99 25.2 8.99 4.28 2.48 6.72 1.44 0.79     4.28 8.99 25.2 8.99 0.79 2.48 6.72 1.44       8.99 4.28 8.99 25.2 1.44 0.79 2.48 6.72   P =  6.72 2.48 0.79 1.44 9.44 3.32 1.78 3.32  ⊗ I2 ,      1.44 6.72 2.48 0.79 3.32 9.44 3.32 1.78     0.79 1.44 6.72 2.48 1.78 3.32 9.44 3.32  2.48 0.79 1.44 6.72 3.32

1.78 3.32 9.44

 8.28 −0.27 −0.8 −0.27 2.9 −0.9 −1.26 −0.74  −0.27 8.28 −0.27 −0.8 −0.74 2.9 −0.9 −1.26     −0.8 −0.27 8.28 −0.27 −1.26 −0.74 2.9 −0.9       −0.27 −0.8 −0.27 8.28 −0.9 −1.26 −0.74 2.9   ⊗ I2 , Q=  2.9 −0.74 −1.26 −0.9 4.94 0.98 0.04 0.98       −0.9 2.9 −0.74 −1.26 0.98 4.94 0.98 0.04     −1.26 −0.9 2.9 −0.74 0.04 0.98 4.94 0.98  −0.74 −1.26 −0.9 2.9 0.98 0.04 0.98 4.94 such that condition (24) holds. Figure 5 shows second-order consensus with information feedback and a coupling delay on the cyclic graph for the case that β = 2 and τ = 0.2. Figures 5(a) and 5(b) plot the curves of the values of xi , which are convergent for x-axis and y-axis, respectively. Figures 5(c) and 5(d) depict the curves of the values of vi , which are convergent for x-axis and y-axis, respectively. 

Fig. 5 Second-order consensus with information feedback and a coupling delay on the cyclic graph (τ = 0.2, β = 2).

5 Conclusions In this paper, we investigate a set of second-order consensus algorithms with coupling delay under general directed information exchange topology. Based on the Lyapunov–Krasovskii functional method, a consensus condition for delaydependent asymptotical stability in terms of LMIs is derived for fundamental second-order consensus algorithm, and

No. 1

Second-Order Consensus of Multiple Agents with Coupling Delay

109

consensus conditions for both delay-independent and delay-dependent asymptotical stabilities in terms of LMIs are derived for the second-order consensus algorithm with information feedback.

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