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Journal of Systems Engineering and Electronics Vol. 25, No. 6, December 2014, pp.1056–1064

Weighted average consensus problem in networks of agents with diverse time-delays

Wenhui Liu1 , Feiqi Deng1,* , Jiarong Liang2 , and Xuekui Yan1

1. College of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, China; 2. School of Computer and Electronic Information, Guangxi University, Nanning 530004, China

Abstract: This paper studies the weighted average consensus problem for networks of agents with ?xed directed asymmetric unbalance information exchange topology. We suppose that the classical distributed consensus protocol is destroyed by diverse time-delays which include communication time-delay and self time-delay. Based on the generalized Nyquist stability criterion and the Gerschgorin disk theorem, some suf?cient conditions for the consensus of multi-agent systems are obtained. And we give the expression of the weighted average consensus value for our consensus protocol. Finally, numerical examples are presented to illustrate the theoretical results. Keywords: networks of agents, distributed control, weighted average consensus, time-delay, digraph theory. DOI: 10.1109/JSEE.2014.00121

1. Introduction

The distributed consensus problem of multi-agent networks has attracted great interests in recent years. The classical consensus protocol has wide applications. It can be used to cope with synchronization control problem [1–3], tracking control problem [4], formation control problem [5], ?ocking control problem [6], and so on. In practice, the environment of multi-agent networks is complicated [7]. Further studying the consensus of multiagent networks with diverse constraint conditions has important theoretical and practical signi?cance [8–16]. Particularly the time-delay is an important factor which has to be considered. We note that the actual network always has diverse time-delays which are induced by many reasons. These delays will generally lead to a reduction of the performance or instability of systems. Therefore, this paManuscript received March 29, 2013. *Corresponding author. This work was supported by the National Natural Science Foundation of China (61273126; 61363002; 61374104), the Natural Science Foundation of Guangdong Province (10251064101000008; S2012010009675), and the Fundamental Research Funds for the Central Universities (2012ZM0059).

per will further study the consensus problem of multi-agent networks with diverse time-delays. The distributed average consensus problem usually means to design a network protocol such that the states of all agents asymptotically reach the average value of their initial states as time goes on. In the 1980s, Tsitsiklis et al. studied the consensus problem and proposed some weighted average protocols [17]. Recently, more and more studies researched the distributed consensus problem of multi-agent networks with diverse time-delays, such as [8– 12,14,15] and references therein. Reference [11] studied the consensus problem for discrete-time multi-agent systems with input and communication delays based on the frequency-domain analysis. For the continuous-time multiagent systems with diverse time-delays, most of the works in the literature assume that each agent has the same self time-delays [8–10]. Therefore, it is necessary and important to investigate the continuous-time multi-agent systems with different self time-delays. The main contributions of this paper are as follows. It is shown that the classical distributed weighted average consensus protocol has a good robustness to diverse timedelays. And the communication time-delays among agents and the self time-delays of each agent have different in?uences on the consensus of multi-agent systems. The remainder of this paper is organized as follows. In Section 2, we propose the distributed protocol with multiple time-delays and formulate the problem to be investigated. Section 3 is the main results. It proves that the multi-agent network can reach weighted average consensus asymptotically. And it gets the consensus value of the network of agents accurately. Simulation results are given in Section 4. Section 5 draws the conclusions. Notations: Rn is the n-dimensional real Euclidean space. C is the 1-dimensional complex space. AT denotes the transpose of A. 1 denotes a column vector with all ones.

Wenhui Liu et al.: Weighted average consensus problem in networks of agents with diverse time-delays

1057

Let G (V , E , A) be a simple weighted digraph with the set of nodes V = {1, . . . , N }, the set of edges E ? V × V and the weighted adjacency matrix A = (aij )N ×N . i (or vi ) denotes the ith node of G . (i, j ) ∈ E denotes a directed edge of G from i to j . aij 0 is the adjacency element such that aij > 0 ? (j, i) ∈ E . The set of in-neighbors {j ∈ V : (j, i) ∈ E}. of the ith node is denoted by Ni

N

di

j =1

aij denotes the in-degree of the ith node. The

τ ?ij and for t 0, i = 1, 2, . . . , N , where 0 τij (t) 0 σij (t) σ ?ij respectively represent the different timedelays at time t. Indeed, when τij (t) = σij (t) ≡ 0, the protocol is just the classical distributed weighted average consensus protocol (2). ?ij 0 and σij (t) ≡ 0, the protocol can When τij (t) ≡ τ be rewritten as the following form [9]: ui (t) = pi

j ∈N i

Laplacian matrix of G is de?ned as L = D ? A, where D = diag{d1 , . . . , dN }. The node is a globally reachable node, if it can reach other nodes along some directed paths in the digraph.

aij (xj (t ? τ ?ij ) ? xi (t)).

(4)

When τij (t) = σij (t) ≡ τ 0, the protocol can be rewritten as the following form [8]: ui (t) = pi

j ∈N i

2. Preliminaries and problem formulation

In this paper, we consider the weighted average consensus problem for a network of agents with the dynamics: x ˙ i (t) = ui (t), t 0, i = 1, 2, . . . , N (1)

aij (xj (t ? τ ) ? xi (t ? τ )).

(5)

When τij (t) ≡ τ ?ij 0 and σij (t) ≡ τ 0, the protocol can be rewritten as the following form [10]: ui (t) = pi

j ∈N i

where xi (t) ∈ R and ui (t) ∈ R are state and input of the ith agent respectively. The communications among agents are modeled as a simple weighted digraph G = (V , E , A), where V = {1, . . . , N } is a set of N nodes, each node denotes an agent, and E ? V × V denotes the edge set. De?nition 1 The network of agents (1) can achieve consensus asymptotically, if lim x(t, x0 ) = α(x0 )1 for every initial condition x0 , where α(x0 ) is a constant. Speci?cally, if α(x0 ) is the weighted average value of the initial condition x0 , the network of agents can achieve weighted average consensus asymptotically. In [8], a classical distributed weighted average consensus protocol was proposed as follows: ui (t) = pi

j ∈N i t→∞

aij (xj (t ? τ ?ij ) ? xi (t ? τ )).

(6)

In this paper, we mainly consider the distributed weighted average consensus protocol as follows: ui (t) = pi

j ∈N i

aij (xj (t ? τij ) ? xi (t ? τi ))

(7)

0 and τi 0 respectively represent where constant τij the communication delay of the information ?ow from the j th agent to the ith agent and self-delay of the ith agent. Compared with the protocols (4)–(6), the protocol (7) allows different self-delays for every agent. We will see that this extension requires some insights into the convergence analysis and calculation of the consensus value.

aij (xj (t) ? xi (t))

(2)

3. Main results

Based on the previous works, we further consider these two improved consensus protocols with multiple timedelays under a ?xed directed weighted information exchange topology. Lemma 1 (see Lemma 2 in [5]) The ?xed simple digraph G has a globally reachable node if and only if 0 is a simple eigenvalue of the Laplacian matrix L of G . Remark 1 If the ?xed simple digraph G has a globally reachable node, then the Laplacian matrix L of G such that L1 = 0 and rank(L) = N ? 1. For the networks of agents mentioned above, we make the following assumption. Assumption 1 G = (V , E , A) is a ?xed simple directed graph (which allows asymmetric and unbalance). And G at least has a globally reachable node. Let L be the Laplacian matrix of G .

for t 0, i = 1, 2, . . . , N , where pi > 0 is the control gain. In actual networks, some small time-delays maybe acceptable, but generally the time-delays will lead to a reduction of the performance or instability of networks. Therefore, it is necessary and important to investigate the timedelay problem of multi-agent systems. Especially, this paper will consider the effects of the communication timedelays among different agents and self time-delays of each agent when investigating the distributed control of multiagent networks. As a natural extension of the previous work, we consider an improved distributed weighted average consensus protocol as follows: ui (t) = pi

j ∈N i

aij (xj (t ? τij (t)) ? xi (t ? σij (t))) (3)

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Journal of Systems Engineering and Electronics Vol. 25, No. 6, December 2014

3.1 Distributed weighted average consensus protocol This subsection considers the distributed protocol (7). For convenience, we denote u ?i (t)

j ∈N i

so the characteristic equation of (9) is det(sI + P L(s)) = 0. (11)

aij (xj (t ? τij ) ? xi (t ? τi ))

(8)

for t 0, i = 1, 2, . . . , N . Without loss of generality, we further assume that A = (aij )N ×N is a row stochastic matrix. That is the in-degree di = 1 for i ∈ V . Indeed, we would normalize the adjacent matrix A such that di = 1 by choosing a new control gain pi . The dynamics of the closed-loop systems are described as x ˙ i (t) = pi u ?i (t), t 0, i = 1, 2, . . . , N. (9)

If the roots of (11) are located in the closed left half complex plane, and the geometric multiplicity of the roots which are located in in the imaginary axis is 1, the network (9) is stable. It is easy to know that s = 0 is the single root of (11). Indeed, by Assumption 1 and Lemma 1, we have rank(P L) = rank(L) = N ? 1 and det(0 · I + P L(0)) =

N

pk

k=1

det(L) = 0.

For convenience, we use x0 to denote the initial condition of the network of agents and suppose that the differential equations has a unique solution x(t) x(t, x0 ) for every initial condition x0 . Remark 2 x0 is just a notation which represents the initial condition of the differential equation (9), which is not necessarily x0 = x(0). We will prove that the network can achieve weighted average consensus asymptotically under some mild conditions. Theorem 1 proves that the network (9) can achieve consensus asymptotically. And we get the weighted average consensus value in Theorem 2. Theorem 1 Suppose Assumption 1 holds. And choose 1 {pk }k=1,2,...,N such that ρ max {pk τk } < . Then 1 k N 2 the network (9) can asymptotically achieve consensus. Proof It is easy and ef?cient to prove Theorem 1 by using a frequency domain approach. For the sake of obtaining the characteristic equation of (9), we assume that the initial conditions are zero. We get sXi (s) = pi

j ∈N i

Next, we will prove that all roots of (11) are located in the opened left half complex plane except s = 0. Let F (s) det(I + G(s)) = 0 and G(s) P L(s)/s. Based on the generalized Nyquist stability criterion [18], the roots of F (s) lie on the opened left half complex plane if and only if the Nyquist characteristic curves of G(ω i) = P L(ω i)/ω i for ω ∈ R ? {0} do not enclose the point ?1 + 0i, where i is the imaginary unit such that i2 = ?1. By the Gersgorin discs theorem, the eigenvalues of G(ω i) are all located in D(ω ), where

N

D(ω )

k=1 N j =1 N

s : s ? pk

e?τ k ω i ωi

pk akj e?τkj ωi ,s ∈ C = ωi e?τ k ω i ωi

N

s : s ? pk

k=1

pk

e?τ k ω i ,s ∈ C ωi

=

aij (e?sτij Xj (s) ? e?sτi Xi (s)) (10)

Dk (ω ).

k=1

where Xi (s) is the Laplace transform of xi (t). Denote X (s) = (X1 (s), . . . , XN (s))T , P = diag {p1 , . . . , pN } and L(s) = {lij (s)}N ×N with ? ??aij e?sτij , j ∈ Ni . lij (s) = e?sτi , i = j ? 0, otherwise It is easy to know that L(0) = L. Using the above notations, (10) can be rewritten as sX (s) = ?P L(s)X (s),

If the points ?a + 0i (a 1) are not in the disks {Dk (ω )}k=1,2,...,N for all ω ∈ R ? {0}, i.e., ? a ? pk e?τ k ω i e?τ k ω i > pk ωi ωi

for ω ∈ R ? {0} and k = 1, 2, . . . , N , then the Nyquist characteristic curves of G(ω i) will not enclose the point of ?1 + 0i. sin(x) Note that 1 for all x ∈ R ? {0} and x a 1 > 2 max {pk τk }, we have

1 k N

Wenhui Liu et al.: Weighted average consensus problem in networks of agents with diverse time-delays

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? a ? pk

e?τ k ω i ωi

2

? pk

e?τ k ω i ωi

2

=

2

By Theorem 1 and the ?nal value theorem, we get

s→0

lim sX (s) = lim x(t) = α1.

t→∞

cos(τk ω ) sin(τk ω ) a + pk ? pk ωi ω pk cos(τk ω ) sin(τk ω ) ? pk ωi ω sin(τk ω ) ω

2

? =

Thus we have

s→0

lim slT P ?1 X (s) (= αlT P ?1 1) = L s + L1 (0) + L2 (0) + · · · s 2! ·

a a ? 2 pk

a(a ? 2pk τk ) > 0.

s→0

lim lT P ?1 ? P

If 2ρ < 1, all roots of F (s) = 0 are located in the opened left half complex plane. Therefore, the network (9) is stable. And by Assumption 1 and Lemma 1, we have lim x(t, x0 ) = α(x0 )1, where α(x0 ) is a constant. Theorem 2 gives the consensus value of the network. Theorem 2 Suppose the conditions in Theorem 1 all hold for the network (9), then lim x(t, x0 ) = α(x0 )1, t→∞ where T l (R(0) + P ?1 x(0)) α(x0 ) lT (P ?1 + L1 (0))1 is the weighted average value of the network of agents, l is a left eigenvector of the Laplacian matrix L associated with eigenvalue 0, x0 is the initial condition of the network, and R(0), P , L1 (0) are presented in the proof. Proof We take the Laplace transform in (9) and obtain sXi (s) ? xi (0) = pi

j ∈N i 0 t→∞

sX (s) + P R(s) + x(0) = ?αlT L1 (0)1 + lT R(0) + lT P ?1 x(0) where Ri (0) = Therefore

j ∈N i 0

aij

?τij

xj (t)dt ?

0 ?τ i

xi (t)dt .

α=

lT (R(0) + P ?1 x(0)) . lT (P ?1 + L1 (0))1

aij (e?sτij Xj (s) ? e?sτi Xi (s)) + aij

j ∈N i 0 ?τ i

pi

?τij

e?s(τij +t) xj (t)dt ? (12)

Remark 3 If the network does not have any communication delays and self-delays, the control gains pi = 1 for i ∈ V , then the consensus value of the network of agents lT x(0) is α(x0 ) = = lT x(0), where we let lT 1 = 1 lT 1 without loss of generality. This is just the consensus value under the classical distributed weighted average consensus protocol. Remark 4 Actually, Theorem 2 can be further described as follows. If Assumption 1 holds and we have known lim x(t, x0 ) = α(x0 )1, then

t→∞

e?s(τi +t) xi (t)dt .

α(x0 ) =

lT (R(0) + P ?1 x(0)) , lT (P ?1 + L1 (0))1

Denote R(s) = (R1 (s), . . . , RN (s))T , Ri (s) =

j ∈N i 0

where l, R(0), P , L1 (0) are the same as Theorem 2. Remark 5 In Theorem 1, the condition max {pk τk }<

1 k N

aij

then (12) can be rewritten as

?τij

e?s(τij +t) xj (t)dt ?

0 ?τ i

e?s(τi +t) xi (t)dt ,

sX (s) = ?P L(s)X (s) + P R(s) + x(0).

(13)

According to Taylor’s series expansion formula, we s2 have L(s) = L + sL1 (0) + L2 (0) + · · ·, where Lk (0) 2! is the k -order derivative of L(s) at s = 0. Speci?cally 1 L1 (0) = {lij (0)}N ×N with ? ?aij τij , j ∈ Ni 1 lij (0) = ?τi , i = j . ? 0, otherwise

1 is suf?cient, but not necessary (see Examples 1, 2 and 2 3). We still emphasize that the condition has a great signi?cance. By the estimations of the self-delays for every agent, we always can choose some appropriate control gains {pk }k=1,2,...,N such that the network of agents can asymptotically achieve consensus. Remark 6 If we do not normalize the adjacent matrix A, Theorem 1 can be redescribed as follows. For the network (9), suppose Assumption 1 holds. If 1 ρ max {pk τk dk } < , then 1 k N 2 lim x(t, x0 ) = lT (R(0) + P ?1 x(0)) lT (P ?1 + L1 (0))1

t→∞

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Journal of Systems Engineering and Electronics Vol. 25, No. 6, December 2014

where l, R(0), P are the same as Theorem 2, L1 (0) = 1 {lij (0)}N ×N with ? ?aij τij , j ∈ Ni 1 lij (0) = ?di τi , i = j . ? 0, otherwise 3.2 Consensus protocol with a state derivative This subsection considers a distributed weighted average consensus protocol with a state derivative: ?i (t) ? qi u ?i (t) ui (t) = pi u pi

j ∈N i

It is easy to know that L(0) = L. Equation (16) can be rewritten as sX (s) = ?(P ? sQ)L(s)X (s), so the characteristic equation of (15) is det(sI + (P ? sQ)L(s)) = 0. (17)

If the roots of (17) are located in the closed left half complex plane, and the geometric multiplicity of the roots which are located in the imaginary axis is 1, the network (15) is stable. It is easy to know that s = 0 is the single root of (17). Indeed, by Assumption 1 and Lemma 1, we have rank(L) = N ? 1 and det(0 · I + (P ? 0Q)L(0)) =

N

aij (xj (t ? τij ) ? xi (t ? τi )) ? aij (x ˙ j (t ? τij ) ? x ˙ i (t ? τi ))

j ∈N i

pk (14)

k=1

det(L) = 0.

qi

for t 0, i = 1, 2, . . . , N , where pi > 0 and qi 0 are the control gains. The communications of this network are modeled as a simple weighted digraph G = (V , E , A). The dynamics of the closed-loop systems are described as ?i (t) ? qi u ?i (t) (15) x ˙ i (t) = pi u for t 0, i = 1, 2, . . . , N . Comparing with the protocol (7), we add a state derivative item in the protocol (14). Theorem 3 proves that the network (15) can achieve consensus asymptotically. And we get the consensus value in Theorem 4. Theorem 3 Suppose Assumption 1 holds. And choose 1 {pk , qk }k=1,2,...,N such that ρ = max {pk τk + qk } < . 1 k N 2 Then the network (15) can asymptotically achieve consensus. Proof The proof is similar to Theorem 1, so we will drop some details. We take the Laplace transform (without loss of generality, let the initial conditions be zero in (15)) and get sXi (s) = (pi ? sqi )

j ∈N i

Next, we will prove that all roots of (17) are located in the opened left half complex plane except s = 0. Let F (s) det(I + G(s)) = 0 and G(s) (P ? sQ)L(s)/s. Based on the generalized Nyquist stability criterion [18], the roots of F (s) lie on the opened left half complex plane if and only if the Nyquist characteristic curves of G(ω i) = (P ? ω iQ)L(ω i)/ω i for ω ∈ R ? {0} do not enclose the point ?1 + 0i. By the Gersgorin discs theorem, the eigenvalues of G(ω i) are all located in D(ω ), where

N

D(ω )

k=1 N j =1 N

s : s ∈ C, s ? (pk ? qk ω i) (pk ? qk ω i)akj e?τkj ωi ωi

e?τ k ω i ωi

= e?τ k ω i ωi

s : s ∈ C, s ? (pk ? qk ω i)

k=1

(pk ? qk ω i)

e?τ k ω i ωi

N

=

k=1

Dk (ω ).

aij (e?sτij Xj (s) ? e?sτi Xi (s))

(16)

If the points ?a + 0i (a 1) are not in the disks {Dk (ω )}k=1,2,...,N for all ω ∈ R ? {0}, i.e., ?a ? (pk ? qk ω i) e?τ k ω i e?τ k ω i > (pk ? qk ω i) ωi ωi

where Xi (s) is the Laplace transform of xi (t). Denote X (s) = (X1 (s), . . . , XN (s))T , P = diag {p1 , . . . , pN }, Q = diag{q1 , . . . , qN } and L(s) = {lij (s)}N ×N with ? ??aij e?sτij , j ∈ Ni lij (s) = e?sτi , i = j . ? 0, otherwise

for ω ∈ R ? {0} and k = 1, 2, . . . , N, then the Nyquist characteristic curves of G(ω i) will not enclose the point of ?1 + 0i. sin(x) Note that 1 for all x ∈ R ? {0} and x a 1 > 2 max {pk τk + qk }, we have

1 k N

Wenhui Liu et al.: Weighted average consensus problem in networks of agents with diverse time-delays

1061

?a ? (pk ? qk ω i)

e?τ k ω i ωi

2

? (pk ? qk ω i)

e?τ k ω i ωi

2

= Ri (s) =

j ∈N i 0 ?τ i

0

aij

a ? qk cos(τk ω ) ?

pk sin(τk ω ) + ω i

2

?τij

e?s(τij +t) xj (t)dt ?

pk cos(τk ω ) qk sin(τk ω ) ? ω

?

e?s(τi +t) xi (t)dt , aij (xj (?τij ) ? xi (?τi ))

?i = R

j ∈N i

pk sin(τk ω ) ? qk cos(τk ω ) ? + ω qk sin(τk ω ) ? pk cos(τk ω ) ω i

2

then (18) can be rewritten as sX (s) = x(0) ? (P ? sQ)L(s)X (s) + ?. (P ? sQ)R(s) + QR (19) According to Taylor’s series expansion formula, we have s2 L(s) = L + sL1 (0)+ L2 (0)+ · · ·. Speci?cally L1 (0) = 2! 1 (0)}N ×N with {lij ? ?aij τij , j ∈ Ni 1 lij (0) = ?τi , i = j . ? 0, otherwise By Theorem 3 and the ?nal value theorem, we get

s→0

=

pk sin(τk ω ) a a ? 2qk cos(τk ω ) ? 2 ω a(a ? 2qk ? 2pk τk ) > 0. If 2ρ < 1, all roots of F (s) = 0 are located in the opened left half complex plane. Therefore the network is stable. And by Assumption 1 and Lemma 1, we have lim x(t, x0 ) = α(x0 )1, where α(x0 ) is a constant. Theorem 4 gives the consensus value of the network. Theorem 4 Suppose the conditions in Theorem 3 all hold for the network (15), then lim x(t, x0 ) = α(x0 )1, t→∞ where α(x0 ) ? + P ?1 x(0)) lT (R(0) ? P ?1 QR T ? 1 l (P + L1 (0))1

t→∞

lim sX (s) = lim x(t) = α1.

t→∞

Thus we have

s→0

lim slT P ?1 X (s) (= αlT P ?1 1) = L + s

∞ k=1

s→0

lim lT P ?1 ? (P ? sQ)

sk ? 1 Lk (0) · k!

is the consensus value of the network of agents, l is a left eigenvector of the Laplacian matrix L associated with eigenvalue 0, x0 is the initial condition of the network, ? , P , Q, L1 (0) are presented in the proof. R(0), R Proof We take the Laplace transform in (15) and obtain sXi (s) ? xi (0) = (pi ? sqi )

j ∈N i

? + x(0) = sX (s) + (P ? sQ)R(s) + QR ? + lT P ?1 x(0) ?αlT L1 (0)1 + lT R(0) + lT P ?1 QR where Ri (0) = Therefore α=

j ∈N i 0 0 ?τ i

aij

?τij

xj (t)dt ?

xi (t)dt .

aij e?sτij Xj (s) ? e?sτi Xi (s) +

0

? + P ?1 x(0)) lT (R(0) + P ?1 QR . lT (P ?1 + L1 (0))1

(pi ? sqi )

j ∈N i 0

aij

?τij

e?s(τij +t) xj (t)dt ?

Remark 7 If we do not add a state derivative item to the consensus protocol, i.e., qi = 0 for i ∈ V , the consensus value of the network (15) is α(x0 ) = (18) lT (R(0) + P ?1 x(0)) . lT (P ?1 + L1 (0))1

e

?τ i

?s(τi +t)

xi (t)dt +

qi

j ∈N i

aij (xj (?τij ) ? xi (?τi )).

Denote R(s) = (R1 (s), . . . , RN (s))T ? = (R ?1, . . . , R ? N )T R

This is just the consensus value of the network (9). Remark 8 Actually, Theorem 4 can be further described as follows. If Assumption 1 holds and we have known lim x(t, x0 ) = α(x0 )1, then

t→∞

α(x0 ) =

? + P ?1 x(0)) lT (R(0) + P ?1 QR lT (P ?1 + L1 (0))1

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Journal of Systems Engineering and Electronics Vol. 25, No. 6, December 2014

? , P , Q, L1 (0) are the same as Thewhere l, R(0), R orem 4. Remark 9 For the network (15), the control gain qi is not necessarily non-negative (see Examples 4 and 5). Indeed, Theorem 3 can be further described as follows (for qi ∈ R). For the network (15), suppose Assumption 1 holds, choose {pk , qk }k=1,2,...,N such that ρ max {pk τk +

1 k N

Accordingly the Laplacian matrix of this digraph is 1 ?1 0 ?0 1 0 ? ? 0 ?0.3 1 L=? ?0 0 ? 0.2 ? ?0 0 0 ?1 0 0 ? 0 0 0 1 0 0 ? 0 0 0 ?1 ? ? 0 ?0.7? ?. ?0.8 0 ? ? 1 ?1 ? 0 1

1 |qk |} < , then 2

t→∞

lim x(t, x0 ) =

? + P ?1 x(0)) lT (R(0) + P ?1 QR lT (P ?1 + L1 (0))1

l = (1, 1, 0, 0, 0, 1)T is a left eigenvector of L associated with eigenvalue 0. The communication delays among agents are shown in Fig. 1. And the self-delay of agents is τ = (0.2, 0.5, 0.1, 0.15, 0.3, 0.35)T.

? , P , Q, L1 (0) are the same as Thewhere l, R(0), R orem 4. Remark 10 If we do not normalize the adjacent matrix A, Theorem 3 (and Remark 9) can be redescribed as follows. For the network (15), suppose Assumption 1 holds and max {(pk τk + choose {pk , qk }k=1,2,...,N such that ρ

1 k N

1 |qk |)dk } < , then 2 ? + P ?1 x(0)) l (R(0) + P QR , lim x(t, x0 ) = t→∞ lT (P ?1 + L1 (0))1 ? , P , Q are the same as Theorem 4, where l, R(0), R 1 L1 (0) = {lij (0)}N ×N with ? ?aij τij , j ∈ Ni 1 lij (0) = ?di τi , i = j . ? 0, otherwise Remark 11 By Theorems 1 – 4, we ?nd that the classical distributed weighted average consensus protocol has a good robustness to diverse time-delays. Generally speaking, if the communication delays among agents are ?nite, they will not destroy the consensus of multi-agent network (1). However, if the self-delays of agents are too big, they maybe destroy the consensus. The communication delays among agents and the self-delays of agents all in?uence the consensus value of multi-agent networks.

T ?1

Fig. 1 Directed network topology and communication delays

The initial condition is xi (t) ≡ xi (0) for t ∈ [αi , 0], i = 1, . . . , 6, where [αi , 0] is an appropriate interval and x(0) = (2, 0, ?1, 3, ?2, 1)T. Based on the above conditions, we will choose a set of control gains to illustrate the above theorems. Example 1 Choose p = (2, 0.8, 4, 3, 1, 1)T, then 1 max {pk τk } = 0.45 < . By Theorem 2, the consen1 k 6 2 sus value of the network (9) is α(x0 ) ≈ 0.865 4. We see from Fig. 2 that the consensus value of the states’ evolution curves also is 0.865 4.

4. Numerical examples

We consider a network with 6 agents. The interconnection topology is shown in Fig. 1. And the weight adjacency matrix is ? ? 0 1 0 0 0 0 ?0 0 0 0 0 1? ? ? ?0 0.3 0 0 0 0.7? ? A=? ?0 0 0.2 0 0.8 0 ? . ? ? ?0 0 0 0 0 1? 1 0 0 0 0 0

Fig. 2

Trajectories of the agents’ states in Example 1

Example 2 Choose p = (2, 2, 4, 3, 1, 2)T, then 1 max {pk τk } = 1.0 > . We see from Fig. 3 that the 1 k 6 2 network (9) cannot achieve consensus.

Wenhui Liu et al.: Weighted average consensus problem in networks of agents with diverse time-delays

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Example 5 Choose p = (1.2, 0.6, 1, 0.4, 1.1, 0.8)T and q = ?(0.2, 0.15, 0.35, 0.4, 0.1, 0.2)T, then max

1 k 6

1 {pk τk + |qk |} = 0.48 < . By Theorem 4, the consen2 sus value of the network (15) is α(x0 ) ≈ 0.833 3. We see from Fig. 6 that the consensus value of the states’ evolution curves also is 0.833 3.

Fig. 3

Trajectories of the agents’ states in Example 2

Choose p = (2, 2, 4, 3, 1, 1)T. Although 1 max {pk τk } = 1.0 > , we still see that the network 1 k 6 2 (9) achieves consensus (Fig. 4). By Remark 4, the consensus value of the network (9) is α(x0 ) ≈ 1.216 2. We see from Fig. 4 that the consensus value of the states’ evolution curves also is 1.216 2. Example 3

Fig. 6 Trajectories of the agents’ states in Example 5

Fig. 4

Trajectories of the agents’ states in Example 3

Remark 12 By Theorem 1 and Examples 2 and 3, if 1 max {pk τk } > , we would not determine that whether 1 k 6 2 or not the network of agents can achieve consensus. However, we also emphasize that Theorem 1 has a great signi?cance. Actually, if we have the estimations of the selfdelays, we always can choose some appropriate control gains {pk }k=1,2,...,N such that the network of agents can asymptotically achieves consensus. There is a similarly explanation for Theorem 3, so we will not say it again.

Example 4 Choose p = (1.2, 0.6, 1, 0.4, 1.1, 0.8)T and q = (0.2, 0.15, 0.35, 0.4, 0.1, 0.2)T, then max

1 k 6

5. Conclusions

In this paper, the weighted average consensus problem is considered for directed networks of continuous-time ?rstorder agents under multiple time-delays. It is shown that the classical distributed weighted average consensus protocol has a good robustness to diverse time-delays. Numerical simulations show the effectiveness of the theoretical results. For future research, we hope to relax the constraint conditions for the control gains and time-delays.

{pk τk + qk } = 0.48 <

1 . By Theorem 4, the consen2 sus value of the network (15) is α(x0 ) ≈ 0.925 9. We see from Fig. 5 that the consensus value of the states’ evolution curves also is 0.925 9.

References

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Fig. 5

Trajectories of the agents’ states in Example 4

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Biographies

Wenhui Liu was born in 1983. He received his bachelor degree and master degree from Department of Mathematics, Zhengzhou University, China, in 2007 and 2010, respectively. He is currently a Ph.D. candidate at the College of Automation Science and Engineering, South China University of Technology, China. His research interests include distributed control of multi-agent systems and stability of complex systems. E-mail: liuwenhui52@sina.com Feiqi Deng was born in 1961. He received his Ph.D. degree in control theory and control engineering from South China University of Technology in 1997. Since October 1999, he has been a professor with South China University of Technology and the director of the Systems Engineering Institute of the university. Now he is serving as the chair of the IEEE SMC Guangzhou Chapter, a vice editor-inchief of Journal of South China University of Technology, and a member of the editorial boards of the following journals: Control Theory and Applications, Journal of Systems Engineering and Electronics and Journal of Systems Engineering. His main research interests include stability, stablization, and robust control theory of complex systems, including timedelay systems, nonlinear systems and stochastic systems. E-mail: aufqdeng@scut.edu.cn Jiarong Liang was born in 1966. He received his master degree in mathematics from Central China Normal University. He received his Ph.D. degree in automatic control from South China University of Technology. He is currently working with Guangxi University serving as a professor in the School of Computer and Electronic Information. His main research interests include singular control systems, and computer networking. E-mail: gxuliangjr@163.com Xuekui Yan was born in 1973. He received his B.E. degree in applied mathematics in 1994 and M.S. degree in computational mathematics in 1997 from Hunan University. He is currently working toward his Ph.D. degree in systems engineering at South China University of China. His research interests include systems engineering and pattern recognition. E-mail: 2682166@qq.com

Weighted average consensus problem in networks of agents with diverse time-delays

作者： 作者单位： Wenhui Liu， Feiqi Deng， Jiarong Liang， Xuekui Yan Wenhui Liu,Feiqi Deng,Xuekui Yan(College of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, China)， Jiarong Liang(School of Computer and Electronic Information, Guangxi University, Nanning 530004, China) 系统工程与电子技术（英文版） Journal of Systems Engineering and Electronics 2014(6)

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引用本文格式：Wenhui Liu.Feiqi Deng.Jiarong Liang.Xuekui Yan Weighted average consensus problem in networks of agents with diverse time-delays[期刊论文]-系统工程与电子技术（英文版） 2014(6)