Robust consensus of multi-agent systems with diverse ... - CiteSeerX

Robust consensus of multi-agent systems with diverse input delays and nonsymmetric interconnection perturbations ? Yu-Ping Tian ∗ , Cheng-Lin Liu School of Automation, Southeast University, Nanjing 210096, P.R.China

Abstract The consensus problem of second-order multi-agent systems with diverse input delays is investigated. Based on the frequency domain analysis, decentralized consensus conditions are obtained for the multi-agent system with symmetric coupling weights. Then, the robustness of the symmetric system with nonsymmetric perturbation is studied. A bound of the largest singular value of the perturbation matrix is obtained as the robust consensus condition. Simulation examples illustrate the design procedure of consensus protocols and validate the correctness of the results. Key words: Multi-agent systems, Consensus, Robustness, Input delays.

1

Introduction

The consensus problem, one of the most important and fundamental issues in the coordination control, was initially studied for systems with first-order agents (see, e.g., Vicsek et al. (1995)). It is now clear that under certain connectivity conditions of the agents’ interconnection topology, consensus can always be achieved for firstorder multi-agent systems with fixed or switching topology (Jadbabaie et al., 2003; Ren & Beard, 2005). Recently, it has been also shown that the consensus condition for the first-order multi-agent system is independent of communication delays (Blondel et al., 2005; Wang & Slotine, 2006; Cao et al., 2006; Papachristodoulou & Jadbabaie, 2006). The problem becomes more complicated when consensus protocols are extended to systems of second-order agents. For second-order multi-agent systems, the consensus depends not only on the connectivity of the interconnection topology but also on the parameters of consensus protocols (Ren, 2006). Lee & Spong (2006) presented a general frequency-domain framework to study the consensus problem for systems of high-order agents with non-uniform communication delays. Using the spectral radius theorem, they obtained a frequencydomain consensus condition which is independent of ? This paper was not presented at any IFAC meeting. ∗ Corresponding author. Fax: 86-25-83794974 Email address: [email protected] (Yu-Ping Tian).

Preprint submitted to Automatica

the communication delays. Lin et al. (2007) extends the consensus protocol proposed in Saber & Murray (2004) to the second-order multi-agent systems, in which each agent delays its own measurement of state in the same number as the communication delay so that it could be matched with the delayed states of its neighbors. It is interesting to note that the delay introduced by agents for themselves in the protocols of Saber & Murray (2004) and Lin et al. (2007) as well can be regarded a kind of input delay. Input delay problems have been extensively studied in classic control theory (see, e.g., Niculescu (2001)). For multi-agent systems, input delay can be also understood as the threshold delay, or queuing time, set by each agent. Up to now, however, there are few reports on the consensus problem with input delays, to our knowledge. Some consensus conditions are obtained only for the case of identical delay in Saber & Murray (2004) and Lin et al. (2007). Although the framework proposed in Lee & Spong (2006) can be used to deal with systems with diverse input delays, from the frequency-domain condition given by this framework it is usually difficult (if not impossible) to derive an explicit relationship between delays and system parameters. Moreover, it will be shown later in this paper that the consensus condition given in Lee & Spong (2006) is so conservative that it gives an empty set of available control parameters for the second-order multi-agent system with input delays. Recently, the authors studied the consensus problem for the system of first-order agents with diverse input delays

14 October 2008

and communication delays, and obtained some less conservative consensus conditions which are explicitly dependent on the input delays but independent of the communication delays (Tian & Liu, 2008). In this paper, we further investigate the leader-following consensus algorithm for second-order multi-agent systems with diverse input delays. Based on some early results for the congestion control system with diverse communication delays (Tian, 2005), decentralized conditions with some preconditions are obtained for the multi-agent system with symmetric coupling weights. As it has been shown for the first-order multi-agent system (Tian & Liu, 2008), exploiting the symmetry of system structure allows one to obtain much less conservative consensus condition. However, a very small perturbation may destroy the symmetry. Thus, robustness of multi-agent systems becomes a very important issue, which has been rarely addressed in current literature. In this paper, for the first time we study the robustness of the symmetric systems with nonsymmetric perturbations of coupling weights. A bound of the largest singular value of the nonsymmetric perturbation matrix is obtained by using the robustness analysis method. 2

(V, E, A), i.e., the connection of the nodes in the diagraph G does not change with time. 3 3.1

Consensus under Diverse Input Delays Leader-following consensus algorithm

In the multi-agent systems with n agents, the interconnection topology can be described as a digraph G = (V, E, A). Each agent can be considered as a node in the digraph, and an information flow between two agents can be regarded as a directed path between ewo nodes. Consider a multi-agent system with input delays, each agent of which is described by ξ˙i = ζi , ζ˙i = ui (t − Ti ), i ∈ I,

(1)

where ξi ∈ R, ζi ∈ R, ui ∈ R, and Ti > 0 are the position, velocity, acceleration, and input delay, respectively, of agent i.

Preliminaries of Graph Theory

For system (1), we adopt the leader-following coordination control strategy. Let the dynamics of the leader be determined by

A weighted directed graph (digraph) G = (V, E, A) of order n consists of a set of vertices V = {v1 , ..., vn }, a set of edges E ⊆ V × V and a weighted adjacency matrix A = [aij ] ∈ Rn×n with nonnegative adjacency elements aij . The node indexes belong to a finite index set I = {1, 2, ..., n}. An edge of the weighted diagraph G is denoted by eij = (vi , vj ) ∈ E, i.e., eij is a directed edge from vi to vj . We assume that the adjacency elements associated with the edges of the digraph are positive, i.e., aij > 0 ⇔ eij ∈ E. Moreover, we assume aii = 0 for all i ∈ I. The set of neighbors of node vi is denoted by Ni = {vj ∈ V : (vi , vj ) ∈ E}.

ξ˙0 = ζ0 ,

(2)

where ξ0 ∈ R is the position of the leader, and ζ0 ∈ R is a constant which represents the desired velocity for all the agents. Then, the consensus protocol for the first-order multiagent system (Saber & Murray, 2004) can be easily extended to the leader-following system as follows

In the weighted digraph G = (V, E,P A), the out-degree n of node i is defined as: degout (vi ) = j=1 aij . Let D be the diagonal matrix with the out-degree of each node along the diagonal and call it the degree matrix of G. The Laplacian matrix of the weighted digraph is defined as L = D − A.

ui = ki

µ X

³ ´ aij (ζj − ζi ) + γ(ξj − ξi )

vj ∈Ni

´¶ +bi (ζ0 − ζi ) + γ(ξ0 − ξi ) , i ∈ I, ³

(3)

where ki > 0 and γ > 0, Ni denotes the neighbors of agent i, aij > 0 is the adjacency element of A in the digraph G = (V, E, A), and bi is the linking weight from agent i to the leader (2). Note that bi > 0 if there is a directed edge from agent i to the leader; otherwise, bi = 0. In the rest of the paper we use the notation B = diag{bi , i ∈ I}.

If there is a path in G from node vj to another node vi , then vi is said to be reachable from vj . If node vi is reachable from every other node in the digraph, then it is said to be globally reachable. A digraph is strongly connected if every two of its nodes, say v and u, are such that v is reachable from u and u is reachable from v. Lemma 1.(Lin et al., 2005) Zero is a simple eigenvalue of L, if and only if the digraph G = (V, E, A) has a globally reachable node.

Remark 1. Protocol (3) can be used only for following a leader with constant velocity, or a leader with velocity which is time-varying but asymptotically approaching to a constant. For a leader with time-varying velocity, a local stabilizer ci ζi should be added in the protocol

In this paper, we just consider static topology G =

2

(3). But the analysis of the stability of such a modified protocol will be beyond the scope of this paper.

Then, from (4) it follows that ξ˙ i = ζ i , µ X ³ ˙ ζ i = ki aij (ζ j (t − Ti ) − ζi (t − Ti ))

With consensus protocol (3), the closed-loop form of system (1) is given by

vj ∈Ni

´ +γ(ξ j (t − Ti ) − ξ i (t − Ti )) ´¶ ³ −bi ζ i (t − Ti ) + γξ i (t − Ti ) , i ∈ I.

ξ˙i = ζi , µ X ³ ˙ζi = ki aij (ζj (t − Ti ) − ζi (t − Ti )) vj ∈Ni

´ +γ(ξj (t − Ti ) − ξi (t − Ti )) ³ +bi (ζ0 − ζi (t − Ti )) ´¶ +γ(ξ0 (t − Ti ) − ξi (t − Ti )) , i ∈ I.

(5)

Taking the Laplace transform of (5), we get (4)

sξ i (s) = ζ i (s), µ X ³ aij (ζ j (s)e−sTi − ζi (s)e−sTi ) sζ i (s) = ki vj ∈Ni

´ +γ(ξ j (s)e−sTi − ξ i (s)e−sTi ) ³ ´¶ −sTi −sTi −bi ζ i (s)e + γξ i (s)e . i∈I

Before going to next subsection we give the following lemma. Lemma 2. Assume the interconnection topology graph of n agents together with the leader in system (4) has the leader as a globally reachable node. Then, the matrix L + B has no zero eigenvalues, where L is the Laplacian matrix of the interconnection topology of n agents without leader.

Denote ´ −sTi a vj ∈Ni ij (s + γ)e ³P ´ Hi (s) = . −sTi s2 + ki vj ∈Ni aij (s + γ)e ki

Proof. Consider the interconnection topology graph with n + 1 nodes corresponding to the n agents of system (4) and the leader. Obviously the Laplacian matrix of this topology is given by " ˆ= L

# 0 0 , −ˆb L + B

0 0 ˆ −b L + B

#

" →

0

³P

(7)

Then, using the framework of Lee & Spong (2006) one can get a sufficient condition of consensus as |Hi (jω)| < 1, ∀ω > 0, ∀i ∈ I.

(8)

Rewrite (7) as

where ˆb = [b1 , · · · , bn ]T . Since the leader is a globally ˆ = n by reachable node in the graph, we have rank(L) ˆ Lemma 1. Taking elementary column transforms for L by adding all the other columns to the first column as follows "

(6)

0

0 L+B

Hi (s) =

κi Wi (s) , 1 + κi Wi (s)

where κi = ki

# Wi (s) =

,

³P

´ a , and ij vj ∈Ni

(s + γ)e−sTi , i ∈ I. s2

(9)

Then, the condition (8) is equivalent to we get rank(L + B) = n. Lemma 2 is proved.

2 Re[κi Wi (jω)] > −1/2, ∀ω > 0.

3.2

Consensus condition under symmetric coupling weights

(10)

Actually, such a condition never holds for any κi > 0 when γ > 0, Ti > 0.

Let

In the rest of this section we will derive a less conservative consensus condition for system (4) with symmetric coupling weights.

ξ i = ξi − ξ0 , ζ i = ζi − ζ0 , i ∈ I.

3

For all i ∈ I, we denote

Writing (6) in the vector form, we can get the characteristic equation of system (5) as follows

Di = Ti γ, q

√

3Di −Di2 +

ω0 (i) =

(3Di −Di2 )2 +8Di (1−Di ) 2

Ti

³ det s2 I + diag{ki , i ∈ I}

´ ·diag{(s + γ)e−sTi , i ∈ I}(L + B) = 0, (15)

.

As it is shown in Tian (2005), ω0 (i) is the critical point of the frequency response of the Wi (s) from clockwise part to anti-clockwise part.

where L is the Laplacian matrix corresponding to the interconnection topology for all the agents without the leader.

Let ˆi ∈ I be the agent which has the maximal input delay constant Tˆi , i.e.,

Define F (s) = det(s2 I + diag{ki , i ∈ I}diag{(s + γ)e−sTi , i ∈ I}(L + B)). To prove Theorem 1 it suffices to prove that all the zeros of F (s) are in the open left half of the complex plane.

ˆi = arg max Ti .

(11)

i∈I

Letting s = 0, we have F (0) = det(γdiag{ki }(L + B)). Because the interconnection topology composed of the n agents together with the leader has the leader as a globally reachable node, we get F (0) 6= 0 from Lemma 2.

Then we have the following theorem. Theorem 1. Assume system (4) is composed of n agents and a leader with a static interconnection topology that has the leader as a globally reachable node, and the topology graph has symmetric weights, i.e., aij = aji . For each agent the following preconditions are assumed: Di < 0.4495, ∀i ∈ I,

(12)

ω0 (i) ω0 (i) ≤ Tˆi−1 arctan( ), ∀i ∈ I. γ

(13)

−sTi

Now, define p(s) = det(I+diag{ki }diag{ (s+γ)e }(L+ s2 B)). We will prove all the zeros of p(s) are in the open left half of the complex plane. Based on the general Nyquist stability criterion (Desoer & Wang, 1980), all the zeros of p(s) lie in the open left half of the complex plane, −jωTi }(L + B)), if the eigenloci of diag{ki }diag{ (jω+γ)e (jω)2 −jωTi

i.e., λ(diag{ki }diag{ (jω+γ)e }(L + B))), does not (jω)2 enclose the point (−1, j0) for ω ∈ R.

Then, all the agents in the system asymptotically converge to the leader’s state, if −1 ki (GM (2 i )

X

aij + bi ) < 1, i ∈ I,

For the symmetric weights (aij = aji ), we get L + B = (L + B)T from the definition. Hence, based on Lemma 3, we have

(14)

vj ∈Ni

where GM i is the gain margin of the transfer function Wi (s) defined in (9).

³ ´ (jω + γ)e−jωTi λ diag{ki }diag{ }(L + B) 2 (jω) ³ (jω + γ)e−jωTi ´ = λ (L + B)diag{ki }diag{ } (jω)2 q q ³ −1 }(L + B)diag{ k (GM )−1 } = λ diag{ ki (GM i i ) i −jωTi ´ (jω + γ)e ·diag{GM } i (jω)2 q q ´ ³ −1 }(L + B)diag{ k (GM )−1 } ) ∈ ρ diag{ ki (GM i i i ³ −jωTi ´ (jω + γ)e ·Co 0 ∪ {GM } . i (jω)2

Before proving Theorem 1, we present some useful lemmas as follows. Lemma 3. (Vinnicombe, 2000) Let Q ∈ C n×n , Q = Q? > 0 and T = diag{ti , ti ∈ C}. Then λ(QT ) ∈ ρ(Q)Co(0 ∪ {ti }), where λ(·) denotes matrix eigenvalue, ρ(·) matrix spectral radius, and Co(·) convex hull. Lemma 4. (Tian, 2005) Suppose that the conditions (12) and (13) are satisfied for the frequency response of Wi (s) defined by (9) for all i ∈ I. Then, κCo(0 ∪ {GM i Wi (jω), i ∈ I}) does not contain the point (−1, j0) for any given real number κ ∈ [0, 1) and any ω ∈ (−∞, ∞), where GM i is the gain margin of Wi (s).

Since the spectral radius of any matrix is bounded by its largest absolute row sum, it follows from the condition (14) that q q ³ ´ −1 }(L + B)diag{ k (GM )−1 } ρ diag{ ki (GM ) i i i

Now, we give a proof of Theorem 1 as follows.

4

1

2

1.5

5

0

1

Ei

0.5

3 ³

4 Fig. 1. Network of 5 agents and a leader.

´ −1 = ρ diag{ki (GM ) }(L + B) i X X M −1 ≤ max ki (Gi ) (| aij + bi | + |aij |) i∈I

=

vj ∈Ni

−1 max ki (GM (2 i ) i∈I

X

−0.5

E4

E1

vj ∈Ni

−1

0

0.1

0.2

0.3

0.4

0.5

E3

0.6

0.7

γ

aij + bi )

vj ∈Ni

Fig. 2. Choosing parameter γ.

< 1.

Since T2 is the maximal input delays, we have ˆi = 2. Denote Ei = T2−1 arctan( ω0γ(i) ) − ω0 (i), i = 1, 2, 3, 4, 5, and the condition (13) can be represented as Ei > 0, i = 1, 2, 3, 4, 5. With the given input delays, Ei is a function of γ. The curves of Ei on γ are shown in Figure 2.

Therefore, from Lemma 4 we obtain that q ³ −1 }(L + B) (−1, 0) ∈ / ρ diag{ ki (GM i ) q ´ ³ (jω + γ)e−jωTi ´ −1 } Co 0 ∪ {GM } , ·diag{ ki (GM i i ) (jω)2

From Figure 2 we see that the condition Ei > 0, i = 1, 2, 3, 4, 5, holds if

−jωTi

γ ∈ (0, 0.16].

i.e, the eigenloci of diag{ki }diag{ (jω+γ)e }(L + B)) (jω)2 dose not enclose the point (−1, j0) for all ω ∈ R, which implies that the zeros of F (s) are all in the open left half plane. Theorem 1 is thus proved. 2 3.3

E5

E2 0

(17)

According to (16) and (17), we can choose γ = 0.10 to guarantee the conditions (12) and (13). Step 2: Choosing ki .

An illustrative example of protocol design

For the transfer functions

We use the following example to illustrate the design procedure based on Theorem 1.

Wi (s) =

Example 1: Consider a system (4) of 5 agents and one leader described by (2). The interconnection topology is described in Figure 1. Obviously, the leader is globally reachable. Assume the input delays for the agents are: T1 = 0.5(s), T2 = 1.0(s), T3 = 0.7(s), T4 = 0.6(s) and T5 = 0.8(s). The weights of the edges are: a12 = a21 = 0.30, a25 = a52 = 0.70, a13 = a31 = 0.10, a34 = a43 = 1.10, a42 = a24 = 0.50, b5 = 1.50.

(s + γ)e−Ti s , i = 1, 2, 3, 4, 5, s2

using Matlab simulator we obtain the inverses of their −1 −1 gain margins as GM ' 0.33, (GM ' 0.67, 1 ) 2 ) M −1 M −1 M −1 (G3 ) ' 0.46, (G4 ) ' 0.39, (G5 ) ' 0.53. From the condition (14), the constraints on ki can be calculated as k1 ∈ (0, 3.788), k2 ∈ (0, 0.498), k3 ∈ (0, 0.906), k4 ∈ (0, 0.801), k5 ∈ (0, 0.651), We choose k1 = 3.4, k2 = k3 = k4 = k5 = 0.4 for the simulation.

In the following, we design parameters γ and ki in the consensus protocol (3) so that the agents converge to the leader’s state asymptotically.

With the parameters chosen above and the initial states generated randomly, the agents in the system (4) asymptotically converge to the leader’s state as shown in Figure 3.

Step 1: Choosing γ. Since there is no other theoretic results to compare with, we test the conservatism of our results by simulation. The procedure is as follows. Setting k2 , k3 , k4 and k5 at our theoretic boundary values as 0.498, 0.906, 0.801 and 0.651 respectively, we increase k1 from our boundary

The condition (12) requires γ ∈ (0, 0.4495/Ti ), ∀i ∈ I, which implies that γ ∈ (0, 0.4495/T2 ) = (0, 0.4495).

(16)

5

Theorem 2. Assume that the nominal part of system (18) , i.e., the system without nonsymmetric weight perturbations δij , converges to the leader’s states asymptotically. Let

6 4

ξi

2 0 −2

−6

³ ´−1 M (s) = s2 I + s2 KD(s)(L + B) s2 KD(s),

Leader Agents

−4 0

50

100

150

200

250

300

350

(19)

400

time/second

where K = diag{ki , i ∈ I} and

1.5 1

D(s) = diag{

ζi

0.5 0 −0.5

Leader Agents

−1 −1.5

0

50

100

150

200

250

300

350

(s + γ)e−sTi , i ∈ I}. s2

Then, the agents in the perturbed system (18) converge to the leader’s states asymptotically, if 400

time/second

σ(∆)σ(M (jω)) < 1, ∀ω ∈ R,

Fig. 3. Positions and velocities of the agents under symmetric weights.

where σ(·) denotes the largest singular value of matrix, and ∆ = {∆ij } is the nonsymmetric perturbation matrix, which is defined as follows

value 3.788 until the system has no consensus. Then we find the computational margin for k1 is k1m = 7.519. Using similar procedures we can obtain the other marginal gains as k2m = 0.797, k3m = 1.088, k4m = 0.954, k5m = 0.773. Note that unlike our theoretic results, these computational margins cannot be used in the consensus protocol simultaneously. 4

P    vj ∈Ni δij , ∆ij = −δij ,    0,

j = i, vj ∈ Ni , otherwise.

Proof. Under the same variable transformation as used in the previous section

Robust Consensus under Nonsymmetric Perturbations

ξ i = ξi − ξ0 , ζ i = ζi − ζ0 , i ∈ I,

The consensus condition given by Theorem 1 depends on the strict symmetry of the Laplacian matrix L. In practice, however, perturbations of coupling weights may occur and destroy the symmetry. In the following, we study the robustness of the consensus protocol against nonsymmetric perturbations.

we get the characteristic equation of system (18) as ³ det s2 I + diag{ki , i ∈ I}

´ ·diag{(s + γ)e−sTi , i ∈ I}(L + B + ∆) = 0.

Suppose the symmetric coupling weights of system (4) are subject to some nonsymmetric perturbations, denoted by δij ∈ R for each one. Then the system becomes

(21)

Since the system (18) without nonsymmetric weight perturbations δij converge to the leader’s states asymptotically, the roots of the characteristic equation (15) all lie in the open left half of the complex plane, i.e., the zeros of det(s2 I + s2 KD(s)(L + B)) lie in the open left half of complex plane, and det(L + B) 6= 0.

ξ˙i = ζi , µ X ³ ˙ζi = ki (aij + δij ) (ζj (t − Ti ) − ζi (t − Ti )) vj ∈Ni

´ +γ(ξj (t − Ti ) − ξi (t − Ti )) ³ +bi (ζ0 − ζi (t − Ti )) ´¶ +γ(ξ0 (t − Ti ) − ξi (t − Ti )) , i ∈ I,

(20)

In the following, we will prove the roots of the equation (21) are all in the open left half of the complex plane. (18)

First we show that Eq. (21) has no roots at s = 0. Indeed, letting ω = 0 we get from (20) that σ(∆)σ((γK(L + B))−1 γK) < 1. This implies that σ[(γK(L + B))−1 γK∆] < 1. So we have

where aij = aji , and aij + δij > 0 hold for vj ∈ Ni . A robust consensus condition of the perturbed system is given by the following theorem.

det(I + (γK(L + B))−1 γK∆) 6= 0,

6

-

d − 6

-

≤ σ(∆)σ(M (jω)) < 1, ∀ω ∈ R.

∆

KD(s)

Hence, we know that λ(∆M (jω)) does not enclose the point (−1, j0) for all ω ∈ R, i.e., the roots of the characteristic equation (23) all lie in the open left half of the complex plane. Therefore, the closed-loop system in Figure 5 is asymptotically stable, and the agents in (18) converge to the leader’s states asymptotically. Theorem 2 is proved. 2

-? d+

L+B

¾

Example 2: Consider the multi-agent systems (18) of 5 agents and one leader described by (2) with the same interconnection topology as Example 1 (Figure 1). For simplicity, we choose the same aij , bi , input delays Ti and control parameters γ and κi , i ∈ I as given in Example 1. From Theorem 1, the system (18) without nonsymmetric weight perturbations converges to the leader’s states asymptotically, and the zeros of det(s2 I + s2 KD(s)(L + B)) lie in the left half of the complex plane. Using the Matlab simulator, we obtain that the largest value of σ(M (jω)) on ω ∈ (−∞, ∞) is maxω∈(−∞,∞) σ(M (jω)) ≈ 23.2. From Theorem 2, if the largest singular value of the nonsymmetric disturbance matrix ∆, i.e., σ(∆), satisfies σ(∆) < (1/23.2), the closed system in Figure 4 with ∆ is asymptotically stable. For example, when

Fig. 4. System with nonsymmetric perturbation.

d − 6

-

∆

M (s)

¾

Fig. 5. Transformed system.

or equivalently, det(γK(L + B + ∆)) 6= 0.



This proves that Eq. (21) has no roots at s = 0. Therefore, the characteristic equation (21) can be equivalently rewritten as ³ ´ det I + KD(s)(L + B + ∆) = 0. (22)

    ∆=   

The feedback diagram corresponding to the characteristic equation (22) is demonstrated in Figure 4. Using the linear fractional transformation, the diagram in Figure 4 can be equivalently transformed into the form shown by Figure 5, where M (s) is given by Eq. (19).

0.005 −0.015 0.01 0

0.02

0

0

0

0

0

−0.015

0

0

−0.02

0

0.01 −0.01

0

0

−0.02 0.02 0

0

0

     ,   

0.015

we have σ(∆) = 0.0375 < (1/23.2), and aij + δij > 0 for vj ∈ Ni . Therefore, with the Laplacian matrix L+∆ and the initial states generated randomly, the agents in (18) converge to the leader’s states asymptotically as shown in Figure 6.

The characteristic equation of the closed-loop system in Figure 5 is det(I + ∆M (s)) = 0.

(24)

5

Conclusion

(23) In this paper, we consider the leader-following consensus algorithm for the second-order multi-agent system with diverse input delays. Sufficient conditions are obtained for the consensus convergence of the multi-agent systems with symmetric coupling weights and nonsymmetric coupling weights respectively. Firstly, we give decentralized consensus conditions for the multi-agent systems with symmetric coupling weights and diverse input delays. Then, we analyze the robustness of the multiagent systems with nonsymmetric perturbations of coupling weights. A bound of the largest singular value of the nonsymmetric perturbation matrix is obtained as the robustness condition for the consensus problem.

Obviously, D(s) has no poles in the open right half of the complex plane. Thus, we know that ∆M (s) has no poles in the open right half of the complex plane. According to the general Nyquist stability criterion (Desoer & Wang, 1980), the roots of the characteristic equation (23) all lie in the open left half of the complex plane, as long as the eigenloci of ∆M (s), i.e., λ(∆M (jω)), does not enclose the point (−1, j0) for ω ∈ R. From condition (20) we have ρ(∆M (jω)) ≤ σ(∆M (jω))

7

ican Control Conference, Minneapolis, Minnesota, pp.756-761. Lin, P., Jia, Y., Du, J. & Yuan, S. (2007). Distributed consensus control for second-order agents with fixed topology and time-delay, Proc. of the 26th Chinese Control Conference, Zhangjiajie, China, pp.577-581. Lin, Z., Francis, B. & Maggiore, M. (2005). Necessary and sufficient graphical conditions for formation control of unicysle. IEEE Trans. on Auto. Control, 50, 121-127. Niculescu, S.-I. (2001). Delay Effects on Stability: A Robust Control Approach, Lecture Notes in Control and Information Sciences, 269, Berlin: Springer. Olfati-Saber, R., & Murray, R. (2004). Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. on Auto. Control, 49, 1520-1533. Papachristodoulou, A. & Jadbabaie, A. (2006). Synchronization in oscillator networks wiht heterogeneous delays, switching topologies and nonlinear dynamics. Proc. of the 45th IEEE Conference on Decision and Control, San Diega, CA, USA, pp.4307-4312. Ren, W. & Beard, R.W. (2005). Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Trans. on Auto. Control, 50, 655-661. Ren, W. (2006). Consensus based formation control strategies for multi-vehicle systems. Proc. of the American Control Conference, Minneapolis, MN, pp.42374242. Tian, Y.-P. (2005). Stability analysis and design of the second-order congestion control for networks with heterogeneous delays. IEEE/ACM Trans. on Networking, 13, 1082-1093. Tian, Y.-P. & Liu, C.-L. (2008). Consensus of multiagent systems with diverse input and communication delays. IEEE Trans. on Auto. Control, in press. Wang, W. & Slotine, J.J.E. (2006). Contraction analysis of time-delayed communication delays. IEEE Trans. on Auto. Control, 51, 712-717. Vicsek, T., Czirok, A., Jacob, E.B., Cohen, I., & Schochet, O. (1995). Novel type of phase transitions in a system of self-driven particles. Phys. Rev. Lett., 75, 1226-1229. Vinnicombe, G. (2000). On the stability of end-to-end congestion control for the internet. Technical report CUED/F-INFENG/TR. No.398, December.

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Although all the consensus conditions given in this paper are obtained based on the frequency domain analysis and thus valid only for fixed network topologies and constant delays, we believe there would be a possible way to extend the results to switching topologies and time-varying delays if a common Lyapunov function or multiple Luapunov functions can be found. Acknowledgement This work was supported by National Natural Science Foundation of China (under grant 60425308), National “863” Programme of China (under grant 2006AA04Z263) and Natural Science Foundation of Jiangsu Province of China (under grant BK2006097). References Blondel, V.D., Hendrickx, J.M., Olshevsky, A. & Tsitsiklis, J. N. (2005). Convergence in multi-agent coordination, consensus, and flocking. Proc. of the 44th IEEE Conference on Decision and Control, and the European Control Conference, Seville, Spain, pp.29963000. Cao, M., Morse, A.S. & Anderson, B.D.O. (2006). Reaching an agreement using delayed information. it Proc. of the 45th IEEE Conference on Decision and Control, San Diega, CA, USA, pp.3375-3380. Desoer, C.A. & Wang, Y.T. (1980). On the generalized Nyquist stability criterion. IEEE Trans. on Auto. Control, 25, 187-196. Jadbabaie, A., Lin, J., & Morse, A. S. (2003). Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. on Auto. Control, 48, 988-1001. Lee, D., Spong, M.W. (2006). Agreement with nonuniform information delays, Proceedings of the Amer-

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