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Finite-time Consensus of Heterogeneous Multi-agent Systems with Linear and Nonlinear Dynamics
Finite-time Consensus of Heterogeneous Multi-agent Systems with Linear and Nonlinear Dynamics ZHU Ya-Kun1
GUAN Xin-Ping2, 3
LUO Xiao-Yuan2
Abstract In this paper, the finite-time consensus problems of heterogeneous multi-agent systems composed of both linear and nonlinear dynamics agents are investigated. Nonlinear consensus protocols are proposed for the heterogeneous multi-agent systems. Some sufficient conditions for the finite-time consensus are established in the leaderless and leader-following cases. The results are also extended to the case where the communication topology is directed and satisfies a detailed balance condition on coupling weights. At last, some simulation results are given to illustrate the effectiveness of the obtained theoretical results. Key words Finite-time consensus, heterogeneous multi-agent systems, linear and nonlinear dynamics Citation Zhu Ya-Kun, Guan Xin-Ping, Luo Xiao-Yuan. Finite-time consensus of heterogeneous multi-agent systems with linear and nonlinear dynamics. Acta Automatica Sinica, 2014, 40(11): 2618−2624
Distributed control of networked multi-agent systems is an important research field due to its important role in a number of applications, such as the formation control of multiple robotics, attitude alignment of satellite clusters, cooperative control of unmanned aerial vehicles, target tracking of sensor networks, and so on[1−5] . The consensus of multi-agent systems means that the states of all the agents converge to a common value by applying effective consensus protocols. The convergence speed is an important factor to evaluate the performance of consensus protocols of multi-agent systems. Lots of researchers found that the convergence rates can be influenced by the second smallest eigenvalue of the interaction graph Laplacian. Some results have been developed on how to enhance the convergence speed of multi-agent system in recent years. Reference [6] found that the second smallest eigenvalue of a regular network can be increased greatly via changing the inter-agent information flow without increasing the total number of the network links. Reference [7] proposed the asymptotic and per-step convergence factors as measures of the convergence speed. One can increase convergence speed with respect to the linear protocols by maximizing the algebraic connectivity of interaction graph, but the consensus can never be reached in a finite time. However, in many practical applications, consensus can be achieved in a finite time is often required, such as when high precision performance and strict convergence time are required, when the control accuracy is crucial, etc. Reference [8] proved that if the interaction topology of a multi-agent system is connected in a sufficiently large time intervals, the proposed protocols can solve the finite-time consensus problems. Reference [9] studied the finite-time consensus of a multi-agent system using a binary consensus protocol. However, all the aforementioned multi-agent systems were homogeneous, which means that all the agents have the same dynamics. In practical systems, the dynamics of agents may be quite different because of various restricManuscript received June 17, 2013; accepted October 8, 2013 Supported by National Basic Research Program of China (973 Program) (2010CB731800), National Natural Science Foundation of China (60934003, 61074065), Key Project for Natural Science Research of Hebei Education Department (ZD200908), and the Doctor Foundation of Northeastern University at Qinhuangdao (XNB201507) Recommended by Associate Editor CHEN Jie 1. School of Control Engineering, Northeastern University (Qinhuangdao), Qinhuangdao 066004, China 2. Institute of Electrical Engineering, Yanshan University, No. 438, Heibei Avenue, Qinhuangdao 066004, China 3. School of Electronic, Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
tions in complicated environments or the different task divisions. For example, in a robot football match, according to the different role tasks, the forward robots and the midfield robots are in charge of shooting, the full back robots are responsible for the defense of particular areas. Each robot in a team has a different function and different dynamics. The team is a system with mixed robots, and can be seen as a heterogeneous multi-agent system. Because of the important application in many practical areas, some researchers start to pay attention to the heterogeneous multiagent systems. Tanner et al.[10] studied the cooperation problems of a kind of heterogeneous mobile robot system composed of a group of unmanned ground vehicle (UGV) and a group of unmanned aerial vehicle (UAV). Consensus of velocity can be achieved based on specific communication protocols. Reference [11] studied the stationary consensus of discrete-time heterogeneous multi-agent systems with communication delays. However, the consensus protocols are convergent asymptotically. It is worthy of noting that the extension of finite-time consensus algorithms from the homogeneous case to the heterogeneous case is nontrivial. Only a few researches have studied the finite-time consensus of heterogeneous multi-agent systems. For example, [12−13] studied the finite-time consensus problem of heterogeneous multi-agent systems composed of first-order and second-order integrator agents. Several classes of nonlinear consensus protocols were constructed. However, all the aforementioned heterogeneous multiagent systems were composed of linear dynamics agents. In reality, nonlinear phenomena widely exist in the mechanical systems, such as teleoperation systems, attitudes of spacecraft and robotic manipulators, and the control accuracy is often crucial. To our best knowledge, there is still no literature concerning the finite-time consensus of heterogeneous multi-agent systems composed of both linear and nonlinear agents. Inspired by the aforementioned discussion, we investigate the finite-time consensus of heterogeneous multi-agent systems composed of linear first-order, second-order integrator agents and nonlinear Euler-Lagrange (EL) agents. The main contributions of this paper can be summarized as follows: 1) We study a kind of heterogeneous multi-agent systems, which is not considered by any existing literature. 2) Some nonlinear heterogeneous control protocols are proposed, such that the heterogeneous multi-agent systems can reach a consensus in a finite time.
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3) It is shown that the investigation of this heterogeneous multi-agent system presents a unified viewpoint to study the finite-time consensus problem. 4) The results are also extended to a class of detailed balance networks and leader-following systems. In addition, linear consensus protocol is mentioned as a special case of the proposed results. The simulations demonstrate that the obtained theoretical results are effective and the heterogeneous multi-agent system has a faster convergence speed under the proposed nonlinear consensus protocol than the corresponding linear protocol. Notation. Throughout this paper, R and R+ stand for the sets of real numbers and positive real number, respectively. Rn denotes the n-dimensional real vector space. Rn×n is the set of n × n matrices. Superscript T stands for the transpose of a matrix or vector. k · k denotes the Euclidean norm. sig(x)α = sgn(x) |x|α , where sgn(·) is the sign function.
1
Preliminaries
1.1
Graph theory
In this subsection, we first review some basic notations of graph theory[14] . For a multi-agent system, agents and information exchange among them are modeled by an undirected weighted graph G = {V, E, A}, where V = {vi |i ∈ Γ } is the set of agents with Γ = {1, 2, · · · , N }, E ⊆ V × V is the set of edges and A is the corresponding weighted adjacency matrix. The adjacency matrix A = [aij ] ∈ Rn×n is defined such that aij > 0 if (vj , vi ) ∈ E, and aij = 0 otherwise. The set of neighbors of agent vi is denoted by Ni = {vj : (vj , vi ) ∈ E}. The of agent vi is defined as deg (vi ) = di = Pn degree P j∈Ni aij . Then the degree matrix of graph j=1 aij = G is D = diag {d1 , · · · , dn } and the Laplacian matrix is L = D − A. A directed graph G is said to be detailed balanced if there exist some scalars ωi > 0 (i ∈ Γ) such that ωi aij = ωj aji for all i, j ∈ Γ. We define one agent as the leader that can only send information to other agents but cannot receive any information from them. 1.2
Problem formulation
Suppose that a heterogeneous multi-agent system consists of linear first-order, second-order integrator agents, and nonlinear EL agents. The number of agents is N . Without loss of generality, we assume that the number of first-order integrator agents is l (l < N ), labeled from 1 to l, the second-order agents are labeled from l + 1 to m, and the EL agents are labeled from m + 1 to N . The linear first-order, second-order and nonlinear EL agents are respectively described as follows: x˙ i (t) = ui (t) , ½ ½
x˙ i (t) = vi (t) , v˙ i (t) = ui (t)
i ∈ Γl
(1)
i ∈ Γm \Γl
(2)
x˙ i (t) = vi (t) , i ∈ ΓN \Γm Mi (xi ) v˙ i + Ci (xi , vi ) vi = ui (t)
(3)
where xi ∈ Rn , vi ∈ Rn and ui ∈ Rn are respectively the position, velocity and control input of agent i, and Γl = {1, 2, · · · , l}, Γm = {1, 2, · · · , m}, ΓN = {1, 2, · · · , N }. The objective of this paper is to design consensus protocols for the heterogeneous multi-agent system (1) ∼ (3) such that the state consensus can be achieved in a finite time.
1.3
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Key lemmas and properties
In this subsection, some important lemmas and properties are given. Lemma 1[15] . Suppose function ϕ : R2 → R satisfies ϕ (xi , xj ) = −ϕ (xj , xi ), i 6= j, i, j ∈ ΓN . Then for any undirected graph G and a set of numbers y1 , y2 , · · · , yN , one has N X X i=1 j∈Ni
−
1 2
aij yi ϕ (xj , xi ) = X
aij (yj − yi ) ϕ (xj , xi )
(vi , vj )∈E
Lemma 2[16] (LaSalle0 s invariance principle). Let Ω be a compact set such that every solution to the system x˙ = f (x), x (0) = x0 starting in Ω remains in Ω for all t ≥ t0 . Let V : Ω → R be a time independent locally Lipschitz and regular function such that D+ V (x (t)) ≤ 0, where D+ denotes the upper Dini derivative. Define © ª S = x ∈ Ω : D+ V (x) = 0 Then every trajectory in Ω converges to the largest invariant set, M , in the closure of S. Next, the homogeneity with dilation is given for the finite-time convergence analysis. For details, one can refer to [17]. Consider the autonomous system x˙ = f (x)
(4)
n
where f : D → R is a continuous function with D ⊂ Rn . A vector field f (x) = (f1 (x) , f2 (x) , · · · , fn (x))T is homogeneous of degree κ ∈ R with dilation r = (r1 , r2 , · · · , rn ), ri > 0 (i ∈ Γn = {1, · · · , n}), if, for any ε > 0, fi (εr1 x1 , εr2 x2 , · · · , εrn xn ) = εκ+ri fi (x) ,
i ∈ Γn
Lemma 3[18] . Suppose that system (4) is homogeneous of degree κ ∈ R with the dilation (r1 , r2 , · · · , rn ), the function f (x) is continuous and x = 0 is asymptotically stable. If the homogeneity degree κ < 0, then system (4) is finite-time stable. Remark 1. Similar to the analysis of [19], one can easily see that the heterogeneous multi-agent system (1) ∼ (3) can achieve consensus in a finite time, if the system (1) ∼ (3) with (x1 , · · · , xN , vl+1 , · · · , vN ) is homogeneous of degree κ < 0 with dilation (R1 , · · · , R1 , R2 , · · · , R2 ) and {z } | {z } | N
N −l
can achieve consensus asymptotically. We revisit the well-known properties of EL systems as follows. One can refer to [20] for more details. Property 1. The inertia matrix M (x) is a symmetric positive-definite function, and there exist positive constants m1 and m2 such that m1 I ≤ M (x) ≤ m2 I. Property 2. The matrix M˙ (x) − 2C (x, v) is skewsymmetric. If matrix A satisfies A = −AT , it is called skew symmetric. Property 2 means that for an arbitrary vector g, ³ ´ g T M˙ (x) − 2C (x, v) g = 0 Property 3. There exist positive scalars Kc such that kC (x, v)k ≤ Kc kvk.
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Main results
2.1
Vol. 40
From Lemma 1 and Properties 1 ∼ 3, one can obtain that
viT
(1 + α)
N X
aij (xj − xi )α +
j=1
i=l+1 m X
2k2
aij ((xj − xi )α )T x˙ i +
i=1 j=1
Ã
m X
In this section, we first propose a finite-time consensus protocol for the heterogeneous multi-agent system (1) ∼ (3) as follows:
N X
Ã
viT
(1 + α)
N X
i=m+1
aij (xj − xi )α +
j=1
N X
2k3
aij sig(vj − vi )β +
j=l+1
N P k1 aij (xj − xi )α , i ∈ Γl j=1 N 1+α P aij (xj − xi )α + 2 j=1 m P k2 aij sig(vj − vi )β , i ∈ Γm \Γl ui = j=l+1 N 1+α P aij (xj − xi )α + 2 j=1 N P aij sig(vj − vi )β , i ∈ ΓN \Γm k3
N X N X
V˙ = − (1 + α)
Finite-time consensus for leaderless case
!
aij sig(vj − vi )β
=
j=m+1
(5)
− (1 + α)
l X N X
aij ((xj − xi )α )T x˙ i −
i=1 j=1 N X
(1 + α)
j=m+1
N X
aij ((xj − xi )α )T vi +
i=l+1 j=1
Ã
m X
viT
(1 + α)
N X j=1
i=l+1
where A = [aij ]N ×N is the aforementioned weighted adjacency matrix associated with graph G; k1 , k2 and k3 are the feedback gains, and they are positive constants; α ∈ (0, 1) is a ratio of odd integers, and β = 2α/(α + 1). With the consensus protocol (5), one can obtain the following theorem. Theorem 1. Suppose graph G is undirected and connected. Then the heterogeneous multi-agent system (1) ∼ (3) reaches consensus in a finite time under protocol (5). Proof. Take a Lyapunov function as V = V1 + V2 + V3 , where
m X
2k2
aij (xj − xi )α +
aij sig(vj − vi )β +
j=l+1 N X
Ã
viT
(1 + α)
N X
aij (xj − xi )α +
j=1
i=m+1 N X
2k3
!
aij sig(vj − vi )β
=
j=m+1
− (1 + α)
l X N X
aij ((xj − xi )α )T x˙ i +
i=1 j=1
V1 =
V2 =
N N Z 1 + α X X xj −xi aij (sα )T ds 2 i=1 j=1 0 m X
viT vi ,
V3 =
N X
m X
2k2
viT
i=l+1 N X
2k3
viT Mi vi
β
aij sig(vj − vi )
j=l+1
Ã
N X
viT
i=m+1
i=m+1
i=l+1
m X
− (1 + α) k1
! β
=
aij sig(vj − vi )
j=m+1
ÃN l X X i=1
+
!T α
aij (xj − xi )
×
j=1
Then one has N X
aij (xj − xi )α −
j=1
V˙ = V˙ 1 + V˙ 2 + V˙ 3 = 1+α 2 2
N X N X
m X i=l+1
k2
aij ((xj − xi )α )T (x˙ j − x˙ i )+ N X i=m+1
viT M˙ i vi + 2
N X i=m+1
k3
(vj − vi )
N X i=m+1
viT Mi v˙ i
m X
T
i=l+1
i=1 j=1
viT v˙ i +
m X
β
aij sig(vj − vi )
j=l+1
à T
(vj − vi )
N X
−
! β
aij sig(vj − vi )
≤0
j=m+1
Thus, by Lemma 2, we have limt→+∞ (xj (t) − xi (t)) = 0 (i, j ∈ ΓN ) and limt→+∞ (vj (t) − vi (t)) = 0 (i, j ∈
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ZHU Ya-Kun et al.: Finite-time Consensus of Heterogeneous Multi-agent Systems with Linear and · · ·
ΓN \Γl ), which means that the origin of the heterogeneous system (1) ∼ (3) with protocol (5) is globally asymptotically stable. Note that system (1) ∼ (3) under protocol (5) is a homogeneous system of degree κ = α − 1 < 0 with dilation (2, · · · , 2, α + 1, · · · , α + 1). Therefore, by Lemma 3, the {z } | {z } | N
N −l
heterogeneous multi-agent system (1) ∼ (3) reaches consensus in a finite time with control protocol (5). Remark 2. Note that the heterogeneous multi-agent system (1) ∼ (3) is a linear heterogeneous multi-agent system when N = m, which has been discussed under an undirected connected communication graph in [12−13]. When l = m = N , system (1) ∼ (3) is a first-order multiagent system, which is a special case of multi-agent system studied in [15]. And when l = 0 and N = m, system (1) ∼ (3) is a second-order multi-agent system investigated in [21]. Therefore, the investigation of the heterogeneous multi-agent system (1) ∼ (3) with the proposed protocol (5) presents a unified viewpoint to solve the finite-time consensus problem of first-order and second-order multi-agent systems, and also the nonlinear EL systems. Remark 3. We can also note that when l = 0, system (1) ∼ (3) is a heterogeneous multi-agent system with linear second-order dynamics agents and nonlinear EL dynamics agents. And when m = l, system (1) ∼ (3) is a heterogeneous multi-agent system with linear first-order dynamics agents and nonlinear EL dynamics agents. To the best of our knowledge, these two kinds of heterogeneous multiagent systems have not been studied. Therefore, the investigation of the heterogeneous multi-agent system (1) ∼ (3) with the proposed protocol (5) presents us a way to discover new researching fields. Remark 4. When α = 1, the nonlinear protocol (5) turns into the corresponding linear protocol (6). N P aij (xj − xi ), i ∈ Γl k1 j=1 N 1+α P aij (xj − xi )+ 2 j=1 m P k2 aij (vj − vi ), i ∈ Γm \Γl ui = j=l+1 N 1+α P aij (xj − xi )+ 2 j=1 N P aij (vj − vi ), i ∈ ΓN \Γm k3
(6)
j=m+1
Under this linear protocol (6), the consensus of the heterogeneous multi-agent system (1) ∼ (3) can only be achieved asymptotically, that is, the heterogeneous multi-agent system (1) ∼ (3) has a faster convergence speed using the nonlinear protocol (5) than using the linear control protocol (6). The network topology studied in Theorem 1 is undirected and connected. As an extension, we consider the directed communication topology in the following. Theorem 2. Suppose graph G is strongly connected and satisfies the detailed balance condition. Then the consensus of heterogeneous multi-agent system (1) ∼ (3) can be achieved in a finite time under the control protocol (5). Proof. Let A˜ = [ωi aij ], and LA˜ be the Laplacian ma˜ From the definition of detail baltrix of topology G(A). ˜ anced graph and the conditions, one can know that G(A) is undirected and connected. Take a Lyapunov function as
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V = V1 + V2 + V3 , where V1 =
V2 =
N N Z 1 + α X X xj −xi ωi aij (sα )T ds 2 i=1 j=1 0 m X
ωi viT vi ,
V3 =
N X
ωi viT Mi vi
i=m+1
i=l+1
Then, similar to the proof of Theorem 1, the consensus of heterogeneous multi-agent system (1) ∼ (3) can be achieved in a finite time under the protocol (5). 2.2 Finite-time consensus for leader-following case In this subsection, we consider the leader-following consensus problem of the heterogeneous multi-agent system (1) ∼ (3). We assume that there is a leader x0 with first-order dynamics, and that only a part of the followers can receive the information from the leader. The consensus protocol for this case is given as follows: N P k1 aij (xj − xi )α + bi (x0 − xi )α , i ∈ Γl j=1 N 1+α P aij (xj − xi )α + bi (x0 − xi )α + 2 j=1 m P k2 aij sig(vj − vi )β , i ∈ Γm \Γl (7) ui = j=l+1 N 1+α P aij (xj − xi )α + bi (x0 − xi )α + 2 j=1 N P aij sig(vj − vi )β , i ∈ ΓN \Γm k3 j=m+1
where bi > 0 when agent i can receive information from leader x0 ; otherwise, bi = 0. Theorem 3. Suppose that the heterogeneous multiagent system (1) ∼ (3) has a leader with first-order dynamics labeled as 0 and at least one of the followers can communicate with it. If the communication topology of the followers is undirected and connected. Then the heterogeneous multi-agent system (1) ∼ (3) can achieve the finitetime consensus under protocol (7). Proof. Let x ¯i = xi − x0 . Choose a Lyapunov function as V = V1 + V2 + V3 + V4 , where N N Z 1 + α X X x¯j −¯xi V1 = aij (sα )T ds 2 i=1 j=1 0
V2 =
m X
viT vi ,
i=l+1
V4 =
N X
V3 =
N X
viT Mi vi
i=m+1
bi x ¯T ¯i i x
i=1
By replacing xi in Theorem 1 with x ¯i , the proof is much similar to that of Theorem 1, and thus is omitted here. For the directed communication topology case, we have the following results. Theorem 4. Suppose that the heterogeneous multiagent system (1) ∼ (3) has a leader with first-order dynamics and at least one of the followers can communicate with it. If the directed communication graph among the followers is strongly connected and satisfies the detailed balance condition, then the heterogeneous multi-agent system (1) ∼ (3) can achieve the finite-time consensus under the control protocol (7).
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Proof. Let x ¯i = xi − x0 . Choose a Lyapunov function as V = V1 + V2 + V3 + V4 , where V1 =
V2 =
N Z N 1 + α X X x¯j −¯xi ωi aij (sα )T ds 2 i=1 j=1 0 m X
ωi viT vi ,
N X
N X
x2 (0) = (0, 2)T , x3 (0) = (2, −1)T , x4 (0) = (−1, 1)T , x5 (0) = (−2, −3)T , x6 (0) = (1, −2)T and the initial velocities are v2 (0) = (−1, 0)T , v3 (0) = (1, 2)T , v4 (0) = (−3, 1)T , v5 (0) = (0, −1)T . For EL agents, we assume that
ωi viT Mi vi
i=m+1
i=l+1
V4 =
V3 =
Vol. 40
· Mi (xi ) =
M11 ∗
M12 M22
¸
· , Ci (xi , vi ) =
C11 C21
¸
ωi bi x ¯T ¯i i x
i=1
Then, the proof is much similar to that of Theorem 1, and thus is omitted here. Remark 5. If there is a leader with second-order dynamic, which is given as follows: ½ x˙ 0 (t) = v0 (t) v˙ 0 (t) = u0 (t) Then protocol (5) turns into the following form. N P k1 aij (xj − xi )α + bi (x0 − xi )α + v0 , i ∈ Γl j=1 N 1+α P aij (xj − xi )α + bi (x0 − xi )α + 2 j=1 m P aij sig(vj − vi )β + k2 j=l+1 ui = bi sig (vj − vi )β , i ∈ Γm \Γl N 1+α P aij (xj − xi )α + bi (x0 − xi )α + 2 j=1 N P aij sig(vj − vi )β + k3 j=m+1 bi sig (vj − vi )β , i ∈ ΓN \Γm (8)
where notation ∗ represents the symmetric part in a symmetric matrix, and xi = (xi1 , xi2 )T , vi = (vi1 , vi2 )T M11 = (2l1 cos xi2 + l2 ) l2 m2 + l12 (m1 + m2 ) M12 =l22 m2 + l1 l2 m2 cos xi2 , C11 = − l1 l2 m2 vi2 sin xi2 ,
M22 = l22 m2 C21 = l1 l2 m2 vi1 sin xi2
C12 = − l1 l2 m2 (vi1 + vi2 ) sin xi2 , m1 =1,
m2 = 0.5,
C22 = 0
l1 = l2 = 0.5
Then, some similar results as in Theorems 3 and 4 can be obtained. Remark 6. When α = 1, these four theorems can still hold with asymptotical convergence speeds.
3
C12 C22
Simulations
In this section, we provide simulations to demonstrate the effectiveness of the obtained theoretical results in this paper. Consider the leaderless heterogeneous multi-agent system (1) ∼ (3) with 6 agents. The communication topology is shown in Fig. 1.
Fig. 1
System topology
Suppose that vertices 1 and 6 denote the first-order integrator agents, vertices 2 and 4 denote the second-order integrator agents, and vertices 3 and 5 denote the nonlinear EL agents. Let k1 = k2 = k3 = 1, α = 3/5. The initial positions of all agents are x1 (0) = (3, 0)T ,
Fig. 2
Simulation results with protocol (5)
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References 1 You Ke-You, Xie Li-Hua. Survey of recent progress in networked control systems. Acta Automatica Sinica, 2013, 39(2): 101−118 (in Chinese) 2 Luo Xiao-Yuan, Shao Shi-Kai, Guan Xin-Ping, Zhao YuanJie. Dynamic generation and control of optimally persistent formation for multi-agent systems. Acta Automatica Sinica, 2012, 38(9): 1431−1438 (in Chinese) 3 Ren W, Atkins E. Distributed multi-vehicle coordinated control via local information exchange. International Journal of Robust and Nonlinear Control, 2007, 17(10−11): 1002−1033 (in Chinese) 4 Sun Wei, Dou Li-Hua, Fang Hao. Cooperative pollution supervising and neutralization with multi-actuator-sensor network. Acta Automatica Sinica, 2011, 37(1): 107−112 5 Olfati-Saber R, Fax J A, Murray R M. Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE, 2007, 95(1): 215−233 6 Nosrati S, Shafiee M. Time-delay dependent stability robustness of small-world protocols for fast distributed consensus seeking. In: Proceedings of the 5th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks and Workshops. Limassol, Cyprus: IEEE, 2007. 1−9 7 Zhou J, Wang Q. Convergence speed in distributed consensus over dynamically switching random networks. Automatica, 2009, 45(6): 1455−1461 8 Wang L, Xiao F. Finite-time consensus problems for networks of dynamic agents. IEEE Transactions on Automatic Control, 2010, 55(4): 950−955
Fig. 3
Simulation results with protocol (6)
Then, using the consensus protocol (5), the state trajectories of all the agents reach consensus as shown in Fig. 2, which verifies the result of Theorem 1. Fig. 3 shows the state trajectories of all agents with linear consensus protocol (6) when α = 1. One can find that the consensus of heterogeneous multi-agent system (1) ∼ (3) can also be achieved. From the comparison of Figs. 2 and 3, we can clearly see that the convergence speed of the heterogeneous multiagent system (1) ∼ (3) under the nonlinear protocol (5) is faster than that under the linear protocol (6). This also illustrates that the proposed nonlinear protocol (5) is effective.
4
Conclusion
In this paper, the finite-time consensus problems of a heterogeneous multi-agent system composed of first-order, second-order and nonlinear Euler-Lagrange agents are investigated. Nonlinear consensus protocols are proposed. Based on the graph theory, LaSalle0 s invariance principle and Lyapunov stability theory, some sufficient conditions of finite-time consensus are established for the leaderless and leader-following multi-agent systems. The results are also extended to the case where the communication topology is directed and satisfies a detailed balance condition on coupling weights. Simulation results are given to demonstrate the effectiveness of the proposed theoretical results.
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19 Zheng Y S, Wang L. Finite-time consensus of heterogeneous multi-agent systems with and without velocity measurements. Systems & Control Letters, 2012, 61(8): 871−878 20 Hua C C, Liu P. A new coordinated slave torque feedback control algorithm for network-based teleoperation systems. IEEE/ASME Transactions on Mechatronics, 2013, 18(2): 764−774 21 Li S H, Du H B, Lin X Z. Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics. Automatica, 2011, 47(8): 1706−1712 ZHU Ya-Kun Lecturer at Northeastern University (Qinhuangdao). He received his Ph. D. degree from the Department of Electrical Engineering, Yanshan University, China in 2013. His research interest covers cooperative control of multi-agent systems and wireless networks. Corresponding author of this paper. E-mail: [email protected]
[email protected]
Vol. 40 GUAN Xin-Ping Professor in control theory and control engineering at Yanshan University and Shanghai Jiao Tong University. He received his M. S. degree in applied mathematics in 1991, and Ph. D. degree in electrical engineering in 1999, both from Harbin Institute of Technology. His research interest covers robust congestion control in communication networks, cooperative control of multi-agent systems, and networked control systems. E-mail:
LUO Xiao-Yuan Professor at Yanshan University. He received his Ph. D. degree from the Department of Electrical Engineering, Yanshan University, China in 2005. His research interest covers multiagent systems, and networked control systems. E-mail: [email protected]
GUAN Xin-Ping2, 3
LUO Xiao-Yuan2
Abstract In this paper, the finite-time consensus problems of heterogeneous multi-agent systems composed of both linear and nonlinear dynamics agents are investigated. Nonlinear consensus protocols are proposed for the heterogeneous multi-agent systems. Some sufficient conditions for the finite-time consensus are established in the leaderless and leader-following cases. The results are also extended to the case where the communication topology is directed and satisfies a detailed balance condition on coupling weights. At last, some simulation results are given to illustrate the effectiveness of the obtained theoretical results. Key words Finite-time consensus, heterogeneous multi-agent systems, linear and nonlinear dynamics Citation Zhu Ya-Kun, Guan Xin-Ping, Luo Xiao-Yuan. Finite-time consensus of heterogeneous multi-agent systems with linear and nonlinear dynamics. Acta Automatica Sinica, 2014, 40(11): 2618−2624
Distributed control of networked multi-agent systems is an important research field due to its important role in a number of applications, such as the formation control of multiple robotics, attitude alignment of satellite clusters, cooperative control of unmanned aerial vehicles, target tracking of sensor networks, and so on[1−5] . The consensus of multi-agent systems means that the states of all the agents converge to a common value by applying effective consensus protocols. The convergence speed is an important factor to evaluate the performance of consensus protocols of multi-agent systems. Lots of researchers found that the convergence rates can be influenced by the second smallest eigenvalue of the interaction graph Laplacian. Some results have been developed on how to enhance the convergence speed of multi-agent system in recent years. Reference [6] found that the second smallest eigenvalue of a regular network can be increased greatly via changing the inter-agent information flow without increasing the total number of the network links. Reference [7] proposed the asymptotic and per-step convergence factors as measures of the convergence speed. One can increase convergence speed with respect to the linear protocols by maximizing the algebraic connectivity of interaction graph, but the consensus can never be reached in a finite time. However, in many practical applications, consensus can be achieved in a finite time is often required, such as when high precision performance and strict convergence time are required, when the control accuracy is crucial, etc. Reference [8] proved that if the interaction topology of a multi-agent system is connected in a sufficiently large time intervals, the proposed protocols can solve the finite-time consensus problems. Reference [9] studied the finite-time consensus of a multi-agent system using a binary consensus protocol. However, all the aforementioned multi-agent systems were homogeneous, which means that all the agents have the same dynamics. In practical systems, the dynamics of agents may be quite different because of various restricManuscript received June 17, 2013; accepted October 8, 2013 Supported by National Basic Research Program of China (973 Program) (2010CB731800), National Natural Science Foundation of China (60934003, 61074065), Key Project for Natural Science Research of Hebei Education Department (ZD200908), and the Doctor Foundation of Northeastern University at Qinhuangdao (XNB201507) Recommended by Associate Editor CHEN Jie 1. School of Control Engineering, Northeastern University (Qinhuangdao), Qinhuangdao 066004, China 2. Institute of Electrical Engineering, Yanshan University, No. 438, Heibei Avenue, Qinhuangdao 066004, China 3. School of Electronic, Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
tions in complicated environments or the different task divisions. For example, in a robot football match, according to the different role tasks, the forward robots and the midfield robots are in charge of shooting, the full back robots are responsible for the defense of particular areas. Each robot in a team has a different function and different dynamics. The team is a system with mixed robots, and can be seen as a heterogeneous multi-agent system. Because of the important application in many practical areas, some researchers start to pay attention to the heterogeneous multiagent systems. Tanner et al.[10] studied the cooperation problems of a kind of heterogeneous mobile robot system composed of a group of unmanned ground vehicle (UGV) and a group of unmanned aerial vehicle (UAV). Consensus of velocity can be achieved based on specific communication protocols. Reference [11] studied the stationary consensus of discrete-time heterogeneous multi-agent systems with communication delays. However, the consensus protocols are convergent asymptotically. It is worthy of noting that the extension of finite-time consensus algorithms from the homogeneous case to the heterogeneous case is nontrivial. Only a few researches have studied the finite-time consensus of heterogeneous multi-agent systems. For example, [12−13] studied the finite-time consensus problem of heterogeneous multi-agent systems composed of first-order and second-order integrator agents. Several classes of nonlinear consensus protocols were constructed. However, all the aforementioned heterogeneous multiagent systems were composed of linear dynamics agents. In reality, nonlinear phenomena widely exist in the mechanical systems, such as teleoperation systems, attitudes of spacecraft and robotic manipulators, and the control accuracy is often crucial. To our best knowledge, there is still no literature concerning the finite-time consensus of heterogeneous multi-agent systems composed of both linear and nonlinear agents. Inspired by the aforementioned discussion, we investigate the finite-time consensus of heterogeneous multi-agent systems composed of linear first-order, second-order integrator agents and nonlinear Euler-Lagrange (EL) agents. The main contributions of this paper can be summarized as follows: 1) We study a kind of heterogeneous multi-agent systems, which is not considered by any existing literature. 2) Some nonlinear heterogeneous control protocols are proposed, such that the heterogeneous multi-agent systems can reach a consensus in a finite time.
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3) It is shown that the investigation of this heterogeneous multi-agent system presents a unified viewpoint to study the finite-time consensus problem. 4) The results are also extended to a class of detailed balance networks and leader-following systems. In addition, linear consensus protocol is mentioned as a special case of the proposed results. The simulations demonstrate that the obtained theoretical results are effective and the heterogeneous multi-agent system has a faster convergence speed under the proposed nonlinear consensus protocol than the corresponding linear protocol. Notation. Throughout this paper, R and R+ stand for the sets of real numbers and positive real number, respectively. Rn denotes the n-dimensional real vector space. Rn×n is the set of n × n matrices. Superscript T stands for the transpose of a matrix or vector. k · k denotes the Euclidean norm. sig(x)α = sgn(x) |x|α , where sgn(·) is the sign function.
1
Preliminaries
1.1
Graph theory
In this subsection, we first review some basic notations of graph theory[14] . For a multi-agent system, agents and information exchange among them are modeled by an undirected weighted graph G = {V, E, A}, where V = {vi |i ∈ Γ } is the set of agents with Γ = {1, 2, · · · , N }, E ⊆ V × V is the set of edges and A is the corresponding weighted adjacency matrix. The adjacency matrix A = [aij ] ∈ Rn×n is defined such that aij > 0 if (vj , vi ) ∈ E, and aij = 0 otherwise. The set of neighbors of agent vi is denoted by Ni = {vj : (vj , vi ) ∈ E}. The of agent vi is defined as deg (vi ) = di = Pn degree P j∈Ni aij . Then the degree matrix of graph j=1 aij = G is D = diag {d1 , · · · , dn } and the Laplacian matrix is L = D − A. A directed graph G is said to be detailed balanced if there exist some scalars ωi > 0 (i ∈ Γ) such that ωi aij = ωj aji for all i, j ∈ Γ. We define one agent as the leader that can only send information to other agents but cannot receive any information from them. 1.2
Problem formulation
Suppose that a heterogeneous multi-agent system consists of linear first-order, second-order integrator agents, and nonlinear EL agents. The number of agents is N . Without loss of generality, we assume that the number of first-order integrator agents is l (l < N ), labeled from 1 to l, the second-order agents are labeled from l + 1 to m, and the EL agents are labeled from m + 1 to N . The linear first-order, second-order and nonlinear EL agents are respectively described as follows: x˙ i (t) = ui (t) , ½ ½
x˙ i (t) = vi (t) , v˙ i (t) = ui (t)
i ∈ Γl
(1)
i ∈ Γm \Γl
(2)
x˙ i (t) = vi (t) , i ∈ ΓN \Γm Mi (xi ) v˙ i + Ci (xi , vi ) vi = ui (t)
(3)
where xi ∈ Rn , vi ∈ Rn and ui ∈ Rn are respectively the position, velocity and control input of agent i, and Γl = {1, 2, · · · , l}, Γm = {1, 2, · · · , m}, ΓN = {1, 2, · · · , N }. The objective of this paper is to design consensus protocols for the heterogeneous multi-agent system (1) ∼ (3) such that the state consensus can be achieved in a finite time.
1.3
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Key lemmas and properties
In this subsection, some important lemmas and properties are given. Lemma 1[15] . Suppose function ϕ : R2 → R satisfies ϕ (xi , xj ) = −ϕ (xj , xi ), i 6= j, i, j ∈ ΓN . Then for any undirected graph G and a set of numbers y1 , y2 , · · · , yN , one has N X X i=1 j∈Ni
−
1 2
aij yi ϕ (xj , xi ) = X
aij (yj − yi ) ϕ (xj , xi )
(vi , vj )∈E
Lemma 2[16] (LaSalle0 s invariance principle). Let Ω be a compact set such that every solution to the system x˙ = f (x), x (0) = x0 starting in Ω remains in Ω for all t ≥ t0 . Let V : Ω → R be a time independent locally Lipschitz and regular function such that D+ V (x (t)) ≤ 0, where D+ denotes the upper Dini derivative. Define © ª S = x ∈ Ω : D+ V (x) = 0 Then every trajectory in Ω converges to the largest invariant set, M , in the closure of S. Next, the homogeneity with dilation is given for the finite-time convergence analysis. For details, one can refer to [17]. Consider the autonomous system x˙ = f (x)
(4)
n
where f : D → R is a continuous function with D ⊂ Rn . A vector field f (x) = (f1 (x) , f2 (x) , · · · , fn (x))T is homogeneous of degree κ ∈ R with dilation r = (r1 , r2 , · · · , rn ), ri > 0 (i ∈ Γn = {1, · · · , n}), if, for any ε > 0, fi (εr1 x1 , εr2 x2 , · · · , εrn xn ) = εκ+ri fi (x) ,
i ∈ Γn
Lemma 3[18] . Suppose that system (4) is homogeneous of degree κ ∈ R with the dilation (r1 , r2 , · · · , rn ), the function f (x) is continuous and x = 0 is asymptotically stable. If the homogeneity degree κ < 0, then system (4) is finite-time stable. Remark 1. Similar to the analysis of [19], one can easily see that the heterogeneous multi-agent system (1) ∼ (3) can achieve consensus in a finite time, if the system (1) ∼ (3) with (x1 , · · · , xN , vl+1 , · · · , vN ) is homogeneous of degree κ < 0 with dilation (R1 , · · · , R1 , R2 , · · · , R2 ) and {z } | {z } | N
N −l
can achieve consensus asymptotically. We revisit the well-known properties of EL systems as follows. One can refer to [20] for more details. Property 1. The inertia matrix M (x) is a symmetric positive-definite function, and there exist positive constants m1 and m2 such that m1 I ≤ M (x) ≤ m2 I. Property 2. The matrix M˙ (x) − 2C (x, v) is skewsymmetric. If matrix A satisfies A = −AT , it is called skew symmetric. Property 2 means that for an arbitrary vector g, ³ ´ g T M˙ (x) − 2C (x, v) g = 0 Property 3. There exist positive scalars Kc such that kC (x, v)k ≤ Kc kvk.
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Main results
2.1
Vol. 40
From Lemma 1 and Properties 1 ∼ 3, one can obtain that
viT
(1 + α)
N X
aij (xj − xi )α +
j=1
i=l+1 m X
2k2
aij ((xj − xi )α )T x˙ i +
i=1 j=1
Ã
m X
In this section, we first propose a finite-time consensus protocol for the heterogeneous multi-agent system (1) ∼ (3) as follows:
N X
Ã
viT
(1 + α)
N X
i=m+1
aij (xj − xi )α +
j=1
N X
2k3
aij sig(vj − vi )β +
j=l+1
N P k1 aij (xj − xi )α , i ∈ Γl j=1 N 1+α P aij (xj − xi )α + 2 j=1 m P k2 aij sig(vj − vi )β , i ∈ Γm \Γl ui = j=l+1 N 1+α P aij (xj − xi )α + 2 j=1 N P aij sig(vj − vi )β , i ∈ ΓN \Γm k3
N X N X
V˙ = − (1 + α)
Finite-time consensus for leaderless case
!
aij sig(vj − vi )β
=
j=m+1
(5)
− (1 + α)
l X N X
aij ((xj − xi )α )T x˙ i −
i=1 j=1 N X
(1 + α)
j=m+1
N X
aij ((xj − xi )α )T vi +
i=l+1 j=1
Ã
m X
viT
(1 + α)
N X j=1
i=l+1
where A = [aij ]N ×N is the aforementioned weighted adjacency matrix associated with graph G; k1 , k2 and k3 are the feedback gains, and they are positive constants; α ∈ (0, 1) is a ratio of odd integers, and β = 2α/(α + 1). With the consensus protocol (5), one can obtain the following theorem. Theorem 1. Suppose graph G is undirected and connected. Then the heterogeneous multi-agent system (1) ∼ (3) reaches consensus in a finite time under protocol (5). Proof. Take a Lyapunov function as V = V1 + V2 + V3 , where
m X
2k2
aij (xj − xi )α +
aij sig(vj − vi )β +
j=l+1 N X
Ã
viT
(1 + α)
N X
aij (xj − xi )α +
j=1
i=m+1 N X
2k3
!
aij sig(vj − vi )β
=
j=m+1
− (1 + α)
l X N X
aij ((xj − xi )α )T x˙ i +
i=1 j=1
V1 =
V2 =
N N Z 1 + α X X xj −xi aij (sα )T ds 2 i=1 j=1 0 m X
viT vi ,
V3 =
N X
m X
2k2
viT
i=l+1 N X
2k3
viT Mi vi
β
aij sig(vj − vi )
j=l+1
Ã
N X
viT
i=m+1
i=m+1
i=l+1
m X
− (1 + α) k1
! β
=
aij sig(vj − vi )
j=m+1
ÃN l X X i=1
+
!T α
aij (xj − xi )
×
j=1
Then one has N X
aij (xj − xi )α −
j=1
V˙ = V˙ 1 + V˙ 2 + V˙ 3 = 1+α 2 2
N X N X
m X i=l+1
k2
aij ((xj − xi )α )T (x˙ j − x˙ i )+ N X i=m+1
viT M˙ i vi + 2
N X i=m+1
k3
(vj − vi )
N X i=m+1
viT Mi v˙ i
m X
T
i=l+1
i=1 j=1
viT v˙ i +
m X
β
aij sig(vj − vi )
j=l+1
à T
(vj − vi )
N X
−
! β
aij sig(vj − vi )
≤0
j=m+1
Thus, by Lemma 2, we have limt→+∞ (xj (t) − xi (t)) = 0 (i, j ∈ ΓN ) and limt→+∞ (vj (t) − vi (t)) = 0 (i, j ∈
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ZHU Ya-Kun et al.: Finite-time Consensus of Heterogeneous Multi-agent Systems with Linear and · · ·
ΓN \Γl ), which means that the origin of the heterogeneous system (1) ∼ (3) with protocol (5) is globally asymptotically stable. Note that system (1) ∼ (3) under protocol (5) is a homogeneous system of degree κ = α − 1 < 0 with dilation (2, · · · , 2, α + 1, · · · , α + 1). Therefore, by Lemma 3, the {z } | {z } | N
N −l
heterogeneous multi-agent system (1) ∼ (3) reaches consensus in a finite time with control protocol (5). Remark 2. Note that the heterogeneous multi-agent system (1) ∼ (3) is a linear heterogeneous multi-agent system when N = m, which has been discussed under an undirected connected communication graph in [12−13]. When l = m = N , system (1) ∼ (3) is a first-order multiagent system, which is a special case of multi-agent system studied in [15]. And when l = 0 and N = m, system (1) ∼ (3) is a second-order multi-agent system investigated in [21]. Therefore, the investigation of the heterogeneous multi-agent system (1) ∼ (3) with the proposed protocol (5) presents a unified viewpoint to solve the finite-time consensus problem of first-order and second-order multi-agent systems, and also the nonlinear EL systems. Remark 3. We can also note that when l = 0, system (1) ∼ (3) is a heterogeneous multi-agent system with linear second-order dynamics agents and nonlinear EL dynamics agents. And when m = l, system (1) ∼ (3) is a heterogeneous multi-agent system with linear first-order dynamics agents and nonlinear EL dynamics agents. To the best of our knowledge, these two kinds of heterogeneous multiagent systems have not been studied. Therefore, the investigation of the heterogeneous multi-agent system (1) ∼ (3) with the proposed protocol (5) presents us a way to discover new researching fields. Remark 4. When α = 1, the nonlinear protocol (5) turns into the corresponding linear protocol (6). N P aij (xj − xi ), i ∈ Γl k1 j=1 N 1+α P aij (xj − xi )+ 2 j=1 m P k2 aij (vj − vi ), i ∈ Γm \Γl ui = j=l+1 N 1+α P aij (xj − xi )+ 2 j=1 N P aij (vj − vi ), i ∈ ΓN \Γm k3
(6)
j=m+1
Under this linear protocol (6), the consensus of the heterogeneous multi-agent system (1) ∼ (3) can only be achieved asymptotically, that is, the heterogeneous multi-agent system (1) ∼ (3) has a faster convergence speed using the nonlinear protocol (5) than using the linear control protocol (6). The network topology studied in Theorem 1 is undirected and connected. As an extension, we consider the directed communication topology in the following. Theorem 2. Suppose graph G is strongly connected and satisfies the detailed balance condition. Then the consensus of heterogeneous multi-agent system (1) ∼ (3) can be achieved in a finite time under the control protocol (5). Proof. Let A˜ = [ωi aij ], and LA˜ be the Laplacian ma˜ From the definition of detail baltrix of topology G(A). ˜ anced graph and the conditions, one can know that G(A) is undirected and connected. Take a Lyapunov function as
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V = V1 + V2 + V3 , where V1 =
V2 =
N N Z 1 + α X X xj −xi ωi aij (sα )T ds 2 i=1 j=1 0 m X
ωi viT vi ,
V3 =
N X
ωi viT Mi vi
i=m+1
i=l+1
Then, similar to the proof of Theorem 1, the consensus of heterogeneous multi-agent system (1) ∼ (3) can be achieved in a finite time under the protocol (5). 2.2 Finite-time consensus for leader-following case In this subsection, we consider the leader-following consensus problem of the heterogeneous multi-agent system (1) ∼ (3). We assume that there is a leader x0 with first-order dynamics, and that only a part of the followers can receive the information from the leader. The consensus protocol for this case is given as follows: N P k1 aij (xj − xi )α + bi (x0 − xi )α , i ∈ Γl j=1 N 1+α P aij (xj − xi )α + bi (x0 − xi )α + 2 j=1 m P k2 aij sig(vj − vi )β , i ∈ Γm \Γl (7) ui = j=l+1 N 1+α P aij (xj − xi )α + bi (x0 − xi )α + 2 j=1 N P aij sig(vj − vi )β , i ∈ ΓN \Γm k3 j=m+1
where bi > 0 when agent i can receive information from leader x0 ; otherwise, bi = 0. Theorem 3. Suppose that the heterogeneous multiagent system (1) ∼ (3) has a leader with first-order dynamics labeled as 0 and at least one of the followers can communicate with it. If the communication topology of the followers is undirected and connected. Then the heterogeneous multi-agent system (1) ∼ (3) can achieve the finitetime consensus under protocol (7). Proof. Let x ¯i = xi − x0 . Choose a Lyapunov function as V = V1 + V2 + V3 + V4 , where N N Z 1 + α X X x¯j −¯xi V1 = aij (sα )T ds 2 i=1 j=1 0
V2 =
m X
viT vi ,
i=l+1
V4 =
N X
V3 =
N X
viT Mi vi
i=m+1
bi x ¯T ¯i i x
i=1
By replacing xi in Theorem 1 with x ¯i , the proof is much similar to that of Theorem 1, and thus is omitted here. For the directed communication topology case, we have the following results. Theorem 4. Suppose that the heterogeneous multiagent system (1) ∼ (3) has a leader with first-order dynamics and at least one of the followers can communicate with it. If the directed communication graph among the followers is strongly connected and satisfies the detailed balance condition, then the heterogeneous multi-agent system (1) ∼ (3) can achieve the finite-time consensus under the control protocol (7).
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Proof. Let x ¯i = xi − x0 . Choose a Lyapunov function as V = V1 + V2 + V3 + V4 , where V1 =
V2 =
N Z N 1 + α X X x¯j −¯xi ωi aij (sα )T ds 2 i=1 j=1 0 m X
ωi viT vi ,
N X
N X
x2 (0) = (0, 2)T , x3 (0) = (2, −1)T , x4 (0) = (−1, 1)T , x5 (0) = (−2, −3)T , x6 (0) = (1, −2)T and the initial velocities are v2 (0) = (−1, 0)T , v3 (0) = (1, 2)T , v4 (0) = (−3, 1)T , v5 (0) = (0, −1)T . For EL agents, we assume that
ωi viT Mi vi
i=m+1
i=l+1
V4 =
V3 =
Vol. 40
· Mi (xi ) =
M11 ∗
M12 M22
¸
· , Ci (xi , vi ) =
C11 C21
¸
ωi bi x ¯T ¯i i x
i=1
Then, the proof is much similar to that of Theorem 1, and thus is omitted here. Remark 5. If there is a leader with second-order dynamic, which is given as follows: ½ x˙ 0 (t) = v0 (t) v˙ 0 (t) = u0 (t) Then protocol (5) turns into the following form. N P k1 aij (xj − xi )α + bi (x0 − xi )α + v0 , i ∈ Γl j=1 N 1+α P aij (xj − xi )α + bi (x0 − xi )α + 2 j=1 m P aij sig(vj − vi )β + k2 j=l+1 ui = bi sig (vj − vi )β , i ∈ Γm \Γl N 1+α P aij (xj − xi )α + bi (x0 − xi )α + 2 j=1 N P aij sig(vj − vi )β + k3 j=m+1 bi sig (vj − vi )β , i ∈ ΓN \Γm (8)
where notation ∗ represents the symmetric part in a symmetric matrix, and xi = (xi1 , xi2 )T , vi = (vi1 , vi2 )T M11 = (2l1 cos xi2 + l2 ) l2 m2 + l12 (m1 + m2 ) M12 =l22 m2 + l1 l2 m2 cos xi2 , C11 = − l1 l2 m2 vi2 sin xi2 ,
M22 = l22 m2 C21 = l1 l2 m2 vi1 sin xi2
C12 = − l1 l2 m2 (vi1 + vi2 ) sin xi2 , m1 =1,
m2 = 0.5,
C22 = 0
l1 = l2 = 0.5
Then, some similar results as in Theorems 3 and 4 can be obtained. Remark 6. When α = 1, these four theorems can still hold with asymptotical convergence speeds.
3
C12 C22
Simulations
In this section, we provide simulations to demonstrate the effectiveness of the obtained theoretical results in this paper. Consider the leaderless heterogeneous multi-agent system (1) ∼ (3) with 6 agents. The communication topology is shown in Fig. 1.
Fig. 1
System topology
Suppose that vertices 1 and 6 denote the first-order integrator agents, vertices 2 and 4 denote the second-order integrator agents, and vertices 3 and 5 denote the nonlinear EL agents. Let k1 = k2 = k3 = 1, α = 3/5. The initial positions of all agents are x1 (0) = (3, 0)T ,
Fig. 2
Simulation results with protocol (5)
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References 1 You Ke-You, Xie Li-Hua. Survey of recent progress in networked control systems. Acta Automatica Sinica, 2013, 39(2): 101−118 (in Chinese) 2 Luo Xiao-Yuan, Shao Shi-Kai, Guan Xin-Ping, Zhao YuanJie. Dynamic generation and control of optimally persistent formation for multi-agent systems. Acta Automatica Sinica, 2012, 38(9): 1431−1438 (in Chinese) 3 Ren W, Atkins E. Distributed multi-vehicle coordinated control via local information exchange. International Journal of Robust and Nonlinear Control, 2007, 17(10−11): 1002−1033 (in Chinese) 4 Sun Wei, Dou Li-Hua, Fang Hao. Cooperative pollution supervising and neutralization with multi-actuator-sensor network. Acta Automatica Sinica, 2011, 37(1): 107−112 5 Olfati-Saber R, Fax J A, Murray R M. Consensus and cooperation in networked multi-agent systems. Proceedings of the IEEE, 2007, 95(1): 215−233 6 Nosrati S, Shafiee M. Time-delay dependent stability robustness of small-world protocols for fast distributed consensus seeking. In: Proceedings of the 5th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks and Workshops. Limassol, Cyprus: IEEE, 2007. 1−9 7 Zhou J, Wang Q. Convergence speed in distributed consensus over dynamically switching random networks. Automatica, 2009, 45(6): 1455−1461 8 Wang L, Xiao F. Finite-time consensus problems for networks of dynamic agents. IEEE Transactions on Automatic Control, 2010, 55(4): 950−955
Fig. 3
Simulation results with protocol (6)
Then, using the consensus protocol (5), the state trajectories of all the agents reach consensus as shown in Fig. 2, which verifies the result of Theorem 1. Fig. 3 shows the state trajectories of all agents with linear consensus protocol (6) when α = 1. One can find that the consensus of heterogeneous multi-agent system (1) ∼ (3) can also be achieved. From the comparison of Figs. 2 and 3, we can clearly see that the convergence speed of the heterogeneous multiagent system (1) ∼ (3) under the nonlinear protocol (5) is faster than that under the linear protocol (6). This also illustrates that the proposed nonlinear protocol (5) is effective.
4
Conclusion
In this paper, the finite-time consensus problems of a heterogeneous multi-agent system composed of first-order, second-order and nonlinear Euler-Lagrange agents are investigated. Nonlinear consensus protocols are proposed. Based on the graph theory, LaSalle0 s invariance principle and Lyapunov stability theory, some sufficient conditions of finite-time consensus are established for the leaderless and leader-following multi-agent systems. The results are also extended to the case where the communication topology is directed and satisfies a detailed balance condition on coupling weights. Simulation results are given to demonstrate the effectiveness of the proposed theoretical results.
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19 Zheng Y S, Wang L. Finite-time consensus of heterogeneous multi-agent systems with and without velocity measurements. Systems & Control Letters, 2012, 61(8): 871−878 20 Hua C C, Liu P. A new coordinated slave torque feedback control algorithm for network-based teleoperation systems. IEEE/ASME Transactions on Mechatronics, 2013, 18(2): 764−774 21 Li S H, Du H B, Lin X Z. Finite-time consensus algorithm for multi-agent systems with double-integrator dynamics. Automatica, 2011, 47(8): 1706−1712 ZHU Ya-Kun Lecturer at Northeastern University (Qinhuangdao). He received his Ph. D. degree from the Department of Electrical Engineering, Yanshan University, China in 2013. His research interest covers cooperative control of multi-agent systems and wireless networks. Corresponding author of this paper. E-mail: [email protected]
[email protected]
Vol. 40 GUAN Xin-Ping Professor in control theory and control engineering at Yanshan University and Shanghai Jiao Tong University. He received his M. S. degree in applied mathematics in 1991, and Ph. D. degree in electrical engineering in 1999, both from Harbin Institute of Technology. His research interest covers robust congestion control in communication networks, cooperative control of multi-agent systems, and networked control systems. E-mail:
LUO Xiao-Yuan Professor at Yanshan University. He received his Ph. D. degree from the Department of Electrical Engineering, Yanshan University, China in 2005. His research interest covers multiagent systems, and networked control systems. E-mail: [email protected]