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Group consensus in multi-agent systems with hybrid protocol
Available online at www.sciencedirect.com
Journal of the Franklin Institute 350 (2013) 575–597 www.elsevier.com/locate/jfranklin
Group consensus in multi-agent systems with hybrid protocol Hong-xiang Hu, Li Yun, Wen-An Zhang, Haiyu Song Department of Automation, Zhejiang University of Technology, Zhejiang Provincial United Key Laboratory of Embedded Systems, Hangzhou 310023, PR China Received 26 February 2011; received in revised form 19 September 2012; accepted 13 December 2012 Available online 15 January 2013
Abstract This paper investigates a group consensus problem with discontinuous information transmissions among different groups of dynamic agents. In the group consensus problem, the agents reach more than one consistent state asymptotically. We consider that the communication topology of these agents, represented by a network, is undirected. Then a novel group consensus protocol, called hybrid protocol, is proposed to solve the couple-group average-consensus problem. The convergence analysis is presented and the algebraic criterions are established. Furthermore, the multi-group consensus is discussed as an extension of the couple-group consensus. By similar techniques, some analysis results are presented. The analysis tools developed in this paper are based on algebraic graph theory, matrix theory, and control theory. Finally, the simulations are provided to demonstrate the effectiveness of the proposed theoretical results. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Recently, distributed coordination of dynamical multi-agent systems has attracted multidisciplinary researchers in various areas including system control theory, statistical physics, biology, applied mathematics, and computer science. One of the most important issue is the agreement or consensus of multi-agent systems, which has been widely applied in many areas such as cooperative control of unmanned air vehicles, formation control, flocking of social insects, wireless sensor networks, mobile robotic swarms, etc. [1–7]. n
Corresponding author. E-mail address: [email protected] (L. Yu).
0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.12.020
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Generally, the main objective of consensus problem is to design an appropriate algorithm or interaction rule such that a group of agents converges to a consistent quantity of interest. The algorithm or the interaction rule is usually called agreement protocol. The consistent quantity that depends on the initial states of all agents is called consensus state which might represent physical quantities such as altitude, position, temperature, voltage, etc. Undoubtedly, the consensus state and the convergence rate are crucial for the consensus problem, which indicate where the multiple dynamic agents reach and how quickly they achieve it, respectively. There have been lots of works studying the consensus problem in the literature. For example, Vicsek et al. [8] presented a discrete-time model containing finite autonomous agents that move in the plane with same speed but different headings, where the concept of neighbors of agents was introduced and some simulations were provided to demonstrate the efficiency of the nearest neighbor rule. Jadbabaie et al. [9] provided a formal analysis for the emergence of alignment in a simplified Viscek’s model of flocking. Olfati-Saber and Murray [10] addressed a systematical framework for consensus problem of dynamic agents with fixed/switching network topology and communication time-delays. In their work, the second smallest eigenvalue of the Laplacian matrix is first used to quantify the convergent speed of the consensus algorithm. More recently, Xiao and Wang [11] studied such asynchronous consensus problem in a continuous-time multi-agent system with discontinuous information transmission by using nonnegative matrix theory and graph theory. Rahmani et al. [12] investigated the controlled agreement problem for multi-agent networks, where several sufficient graphical conditions were obtained based on the graph automorphisms for the system’s uncontrollability. Based on the work introduced in [13,14], Galvan-Guerra et al. [15] studied the specific optimal control problem associated with a multiagent dynamic system. In their work, to minimize the tracking error in the leader-follower model, the hybrid LQ-based optimization techniques were considered as an auxiliary method associated with the constructive solution procedure for the initial multiagent tracking problem. All these results are only concerned with first-order dynamics. Since many practical individual systems, especially the mechanical systems, are of secondorder dynamics, there have been also some related works focusing on the consensus problem of the agents with this kind of dynamics [16–18]. Meanwhile, a variety of methods have been presented in the literature to analyze the convergence rate of the consensus of multi-agent systems. Generally, it is considered that communication topology plays a key role in the convergence of consensus process. For example, Cao et al. presented new graph-theoretic results appropriate for the analysis of a variety of consensus problems cast in dynamically changing environments in [19,20], where the concepts of rooted graph, strongly rooted graph, and neighbor-shared graph were introduced and the worst convergence rates of these graphs were derived. Nedic et al. [21] considered a constrained consensus problem and proposed a distributed projected consensus algorithm with the results of convergence rate being established. Since in some practical situations it may be required that the agreement is reached in finite time, finite-time agreement becomes an important topic over the past few years [22–25]. However, almost all the aforementioned results were only concerned with one common consensus state, which need to be further generalized in several cases. For example, in the formation flight of Unmanned Air Vehicles (UAV), the set of UAV must be divided into sub-sets when some cooperative tasks need to be accomplished synchronously. Therefore, appropriate protocols and algorithms need to be designed such that agents in a network
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reach more than one common consensus state. Fortunately, this problem has been already addressed by Yu and Wang [26,27], which is named as the group consensus problem. In the group consensus problem, the whole network is divided into multiple sub-networks, and all the agents are divided into multiple groups consequently. Moreover, information exchange exists in different groups. Furthermore, it was assumed in [27] that the channels between different groups must exist continuously. Thus, the protocols proposed in [27] are pure continuous-time ones. In practice, the information exchange between different groups may be intermittent due to unreliable communication channels. To the best of our knowledge, the group consensus problem of multi-agent systems with intermittent information exchanges has not yet been investigated in the existing literature, which motivates the present research. In contrast to the protocol presented in [27], this paper investigates a group consensus problem with hybrid protocol. The hybrid protocol includes continuous-time signal that depicts the information exchange in the same group, and discrete-time signal that describes the information exchange between different groups, which is considered more consistent with practical situations. An outline of this paper is as follows. Some basic definitions and results in matrix and graph theories are presented in Section 2. The main results about group consensus and convergence rate are given in Section 3. The results are verified via numerical simulations in Section 4. Finally, the paper is concluded in Section 5. Notations. The notation used in this paper is fairly standard. Specifically, Denote by R and N the sets of real numbers and natural numbers, respectively. Denote by Rn the set of n 1 real column vectors and ( )T the transpose. Meanwhile, N þ ¼ f0g [ N,
1n ¼ ð1,1, 1 ,0 0ÞT 2 Rnþm , |fflfflfflfflffl{zfflfflfflfflffl} n
1m ¼ ð0, 0,1,1, 1 ÞT 2 Rnþm : |fflfflfflfflffl{zfflfflfflfflffl} m
Denote by InARn n the identity matrix and : : the Euclidean vector norm. For AARn n, denote by tr(A), :A:, and r(A) the trace, the norm, and the spectral radius of A, respectively. Specially, denote by :A:F the Frobenius norm. Then we have jjAjj ¼ sup x 2 Rn jjAxjj, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 jjAjjF ¼ i,j ¼ 1 jaij j , and rðAÞ ¼ maxfjli jg with li being the eigenvalue of A.
jjxjj ¼ 1
i
2. Problem formulation Some basic concepts of graph theory and results in matrix theory, which have been used in [28,29], are introduced in this section. In a multi-agent system, each agent can communicate with several other agents which are defined as its neighbors. The topology of such communication relationships then can be represented by a weighted undirected graph G ¼ (V, x, A) of order n with a set of nodes V ¼ fv1 ,v2 ,. . .,vn g, a set of edges xDV V, and a nonnegative symmetric matrix A ¼ [aij]. The node indexes belong to a finite index set I ¼ f1,2,. . .,ng. And the edge of G is denoted by eij ¼ (vi,vj), which is an unordered pair of vertices. (vi,vj)Ax3aij403 agent i and j can communicate with each other, namely, they are adjacent. Moreover, we assume that aii ¼ 0
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for all iAI. Here, A is called the weight matrix and aij is the weight of edge eij ¼ (vi,vj). The set of neighbors of node vi is denoted by Ni ¼ {vj9(vj,vi)Ax}. A path in a graph from vi to vj is a sequence of distinct vertices from vi to vj such that the consecutive vertices are adjacent. A graph is connected if there is at least one path between any two vertices in the graph. Let xiAR denote the state of node vi, we refer to (G,x) with state x ¼ ½x1 ,x2 ,. . .,xn T 2 Rn and topology graph G. Each agent updates its current state based upon the information of its neighbors. Suppose that each agent of a graph has the dynamics as follows: x_ i ðtÞ ¼ ui ðtÞ,
8i 2 I,
ð1Þ
where ui ðtÞ ¼ fi ðxi ,xj1 ,. . .,xjm Þ is a state feedback. If the cluster J ¼ fxj1 ,. . .,xjm gDNi , then ui(t) is said to be a protocol with topology G. Ren and Beard [30] adopted the following protocol to solve the consensus problem and the average consensus problem: X aij ðxj xi Þ, ð2Þ ui ðtÞ ¼ vj 2Ni
It has been proved that if the topology G is undirected and connected, then the protocol (2) asymptotically solves an average consensus problem [10]. Considering practical applications of cooperative control, a more general issue, i.e., the group consensus, is addressed in this paper, where agents in a network reach more than one common consensus state asymptotically. Without loss of generality, we consider the case that agents in a network reach two consistent states asymptotically. Let a complex network (G,x) consists of nþm (n,m41) agents, the group consensus problem is simply defined as follow: the first n agents reach a consistent state while the other m agents reach another consistent state in the presence of information exchange between the two groups. Denote I1 ¼ f1,2,. . .,ng, I2 ¼ fn þ 1,n þ 2,. . .,n þ mg, V1 ¼ fv1 ,v2 ,. . .,vn g, V2 ¼ fvnþ1 ,. . .,vnþm g, N1i ¼ {vjAV19(vj,vi)Ax}, and N2i ¼ {vjAV29(vj,vi)Ax}. Obviously, I¼ I1[I2, V¼ V1[V2, and Ni ¼ N1i[N2i. Then define a new protocol for a complex network (G,x) as follows: P P ( 8i 2 I1 , vj 2N1i aij ðxj xi Þ þ vj 2N2i bij xj ðhk Þ P P ui ðtÞ ¼ , 8t 2 ½hk ,hkþ1 Þ, k 2 N þ , 8i 2 I2 , vj 2N1i bij xj ðhk Þ þ vj 2N2i aij ðxj xi Þ ð3Þ where aijZ0, bijAR, and hk is the update time, which is the time when the information of neighbors from different groups known by agent i is updated. In other words, the update time is the moment when information exchange between different groups takes place. Assume that {hk} satisfies the following conditions: (A1) there exist positive real numbers t0, t1 such that for any kAN, t0rhkþ1hkrt1; (A2) h0 ¼ 0. Since the protocol (3) contains both continuous-time and discrete-time signals, it is called a hybrid protocol. Moreover, the assumptions (A1) and (A2) in [27] are also satisfied
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between different groups, namely (A3) nþm X j ¼ nþ1
bij ¼ 0,
n X
bij ¼ 0:
j¼1
Remark 1. The structure of different groups is shown in Fig. 1, which shows that the protocol (3) can be implemented in the following way: when t¼ hk, the switch is closed, and the information exchange between different groups takes place; when tA(hk,hkþ1), the switch is open, and the information exchange between different groups maintains the value at t ¼ hk due to the zero-order hold. This protocol is different from that introduced in [27], which is a pure continuous-time protocol, that is, channels between different groups must exist continuously. Since in reality the channels between different groups may be unreliable due to disturbances, especially when one group is far away from another, and thus may result in discontinuous information exchanges, the proposed hybrid protocol (3) is more suitable for practical applications than that in [27]. Remark 2. The protocol (3) indicates that an agent may receive the information of agents in another group at the time hk. Meanwhile, one has hk ¼ t(thk), and the term thk could be considered as time-delay, hence the protocol (3), in essence, is a time-delay protocol. However, it is easy to see that different from Eq. (18) in [27], this time-delay is not a continuous function of t, but only a piecewise continuous function. Therefore, the solution of systems (1) and (3) is referred to as sample-and-hold solution [31]. In order to obtain the main results, the following definitions are needed. Definition 1. (Yu and Wang [26]). The hybrid protocol (3) is said to asymptotically solve a couple-group consensus problem, if there exists an x * Aspan{1n, 1m}, such that limt-1 jjxðtÞxn jj ¼ 0, where xðtÞ ¼ ðx1 ,x2 ,.. .,xnþm be the initial xi(0) P PÞ: Furthermore, let state of agent i, if x * ¼ a1nþb1m, where a ¼ 1=nÞ ni ¼ 1 xi ð0Þ and b ¼ 1=mÞ nþm i ¼ nþ1 xi ð0Þ, then it is said that the hybrid protocol (3) asymptotically solves a couple-group averageconsensus problem. Definition 2. (Yu and Wang [27]). A network with topology G1 ¼ (V1, x1, A1) is said to be a sub-network of a network with topology G ¼ (V, x, A) if (i) V1DV, (ii) x1Dx and (iii) the weighted adjacency matrix A1 inherits A. Furthermore, if the inclusion relations in (i) and (ii) are strict, then it is said that G1 is a proper sub-network of G.
ZOH Group 1
Switch ZOH Network
Fig. 1. Multi-agent systems with hybrid protocol.
Group 2
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According to Definition 2, the first n agents constitute a network G1, and the other m agents constitute another network G2. It is assumed that (A4) network G1 and G2 are connected. Before presenting the main results, a useful lemma is introduced as follows. Lemma 1. (Godsil and Royle [28]). Consider an n n symmetrical matrix A ¼ [aij] satisfying Pn a ¼ 0. The following statements hold: ij i¼1 (i) There exists an orthogonal matrix U, such that 0 0 U T AU ¼ , 0 n where * represents any nonzero matrix with an appropriate dimension and 0 represents any zero matrix with an appropriate dimension; (ii) Matrix A has a zero eigenvalue with the eigenvector 1 ¼ ð1,1,. . .,1Þ 2 Rn :
3. Main results In this section, the group consensus problem for multi-agent system (1) with the hybrid protocol (3) is investigated. The convergence analysis is presented and algebraic criterions are established in the couple-group average-consensus problem. Furthermore, the multigroup consensus as an extension of the couple-group consensus is also discussed. Given the protocol (3), the multi-agent system (1) evolves according to the following dynamic: 8 P P aij ðxj xi Þ þ bij xj ðhk Þ 8i 2 I1 , > > < vj 2N1i vj 2N2i P P ð4Þ x_ i ðtÞ ¼ , 8t 2 ½hk ,hkþ1 Þ, k 2 N þ , 8i 2 I2 , bij xj ðhk Þ þ aij ðxj xi Þ > > : vj 2N1i vj 2N2i we use vector notations to rewrite the equations in Eq. (4) with a compact form as follows: 2 3 2 3 2 3 x_ 1 x1 x1 ðhk Þ 6 x_ 7 6 x 7 6 x ðh Þ 7 6 2 7 6 2 7 6 2 k 7 6 6 7 7 6 7 6 ^ 7 7 " #6 ^ 7 " #6 ^ 6 7 6 7 6 7 0 B L 0 1 1 6 x_ 7 6 x 7 6 x ðh Þ 7 6 n 7¼ 6 n 7þ T 6 n k 7, 8t2 ½hk ,hkþ1 Þ, k 2 N þ , B1 0 6 0 L2 6 6 7 7 7 6 x_ nþ1 7 6 xnþ1 7 6 xnþ1 ðhk Þ 7 6 7 6 7 6 7 6 ^ 7 6 ^ 7 6 7 ^ 4 5 4 5 4 5 xnþm ðhk Þ x_ nþm xnþm ð5Þ where L1ARn n and L2ARm m are the Laplacian matrices of network G1 and G2, respectively, and the matrix B1ARn m describes the information exchange between different groups.
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Define
"
L1 L¼ 0
x ¼ ðx1 ,x2 ,. . .,xnþm ÞT ,
# 0 , L2
" B¼
0
B1
BT1
0
581
# ,
and system (5) can be rewritten as _ ¼ LxðtÞ þ Bxðhk Þ, xðtÞ
8t 2 ½hk ,hkþ1 Þ,
k 2 N þ:
ð6Þ
By the theory of ordinary differential equations [32], the solution of Eq. (6) in tA[hk,hkþ1) is Z t eLðttÞ Bxðhk Þ dt xðtÞ ¼ eLðthk Þ xðhk Þ þ hk Z t Lðthk Þ LðttÞ ¼ e þ e B dt xðhk Þ, 8t 2 ½hk ,hkþ1 Þ: ð7Þ hk
When t¼ hkþ1, by the continuity of the solution, we can obtain that Z hkþ1 xðhkþ1 Þ ¼ eLðhkþ1 hk Þ þ eLðhkþ1 tÞ B dt xðhk Þ:
ð8Þ
hk
Define x(hkþ1)¼ x(kþ1), x(hk)¼ x(k) Z hkþ1 CðkÞ ¼ eLðhkþ1 hk Þ þ eLðhkþ1 tÞ B dt hk
and rewrite Eq. (8) as xðk þ 1Þ ¼ CðkÞxðkÞ:
ð9Þ
The following lemma claims that the study of system (9) is sufficient for the investigation of system (6). Lemma 2. System (6) solves the couple-group average-consensus problem if and only if system (9) solves the couple-group average-consensus problem. Proof. The necessity follows from the definition of the state variable x(k). To prove the sufficiency, assume that system (9) solves the couple-group averageconsensus problem. P P Let x * ¼ aU1nþb1m, where a ¼ 1=n ni ¼ 1 xi ð0Þ, b ¼ 1=m nþm i ¼ nþ1 xi ð0Þ such that limk-1 x ðkÞ ¼ xn . Obviously, one has Z t eLðthk Þ þ eLðttÞ B dt xn ¼ xn , CðkÞxn ¼ xn , hk
for any ðk,tÞ 2 N þ ½hk ,hkþ1 Þ: In fact, by the definition of C(k), it follows: Z hkþ1 CðkÞxn ¼ ½eLðhkþ1 hk Þ þ eLðhkþ1 tÞ B dtxn hk
¼e
Lðhkþ1 hk Þ n
Z
hkþ1
x þ hk
eLðhkþ1 tÞ Bxn dt:
ð10Þ
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By the definition of B and (A3), one obtains that Bx * ¼ 0, hence, Z hkþ1 eLðhkþ1 tÞ Bxn dt ¼ 0: hk
Meanwhile, consider the term eLðhkþ1 hk Þ xn , it can be shown that " # 1 X 1 k k k Lðhkþ1 hk Þ n ð1Þ ðhkþ1 hk Þ L xn : x ¼ e k! k¼0
ð11Þ
Due to the fact that Lx * ¼ 0, one obtains that eLðhkþ1 hk Þ xn ¼ xn . Therefore, C(k)x * ¼ x * , and one also has Z t Lðthk Þ LðttÞ e þ e B dt xn ¼ xn : hk
For any given t40, there exists k, such that tA[hk,hkþ1), and Z t Lðth Þ n LðttÞ n k þ e B dt xðhk Þx jjxðtÞx jj ¼ e h Z kt Lðth Þ k ¼ e þ eLðttÞ B dt ðxðhk Þxn Þ h Z tk Lðth Þ k re þ eLðttÞ B dtjjUjjxðhk Þxn :
ð12Þ
hk
Because hkþ1 hk 2 ½t0 ,t1 , 8k 2 N þ , the norm of Z t eLðthk Þ þ eLðttÞ B dt hk
is bounded, namely, there exists a constant M40, such that Z t eLðttÞ B dtjjrM, 8k 2 N þ : jjeLðthk Þ þ hk
Since k-N as t-N, one has lim jjxðtÞxn jjrMU lim jjxðhk Þxn jj ¼ 0,
t-1
k-1
ð13Þ
which yields lim jjxðtÞxn jj ¼ 0:
t-1
ð14Þ
Thus, system (6) solves the couple-group average-consensus problem. This completes the proof. & By Lemma 2, system (9) can be regarded as a discrete-time multi-agent system consisting of nþm agents with time-varying parameters, which should be emphatically investigated.
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Let U1T ¼ ½u11 ,u12 2 Rnn and U2T ¼ ½u21 ,u22 2 Rmm be two orthogonal matrices, such that 2 3 2 3 0 0 6 7 6 7 lnþ2 l2 6 7 6 7 T T 6 7, ð15Þ U1 UL1 UU1 ¼ 6 7, U2 UL2 UU2 ¼ 6 7 & & 4 5 4 5 lnþm ln where 0ol2 rl3 r rln are the eigenvalues of L1, while 0olnþ2 rlnþ3 r rlnþm are the eigenvalues of L2, u11ARn 1, and u21ARm 1. According to Eq. (15), one has u11 ¼ pffiffiffiffi pffiffiffiffi pffiffiffi pffiffiffi ½1= n,. . .,1= nT , u21 ¼ ½1= m, ,1= mT , furthermore, u21ARn (n1), and each column vector of u21 represents an eigenvector of L1, while u22ARm (m1), and each column vector of u21 represents an eigenvector of L2. It is inferred from Eq. (15) that " # " #T " # " # " #T U1 0 U1 0 U1 0 L1 0 U1 0 ¼ ULU U U 0 U2 0 U2 0 U2 0 L2 0 U2 3 2 0 7 6 l2 7 6 7 6 7 6 & 7 6 7 6 ln 7 6 7: 6 ð16Þ ¼6 7 0 7 6 7 6 lnþ2 7 6 7 6 7 6 & 5 4 lnþm Denote " yðkÞ ¼
U1
0
0
U2
# xðkÞ,
then system (9) can be rewritten as yðk þ 1Þ ¼ C 0 ðkÞyðkÞ,
ð17Þ
where "
U1 C ðkÞ ¼ 0 0
# " 0 U1 UCðkÞU U2 0
0 U2
#T :
0
The following lemma claims that C (k) is a symmetric matrix on a joining condition. 0
Lemma 3. Given system (17), if L1UB1 ¼ B1UL2, then C (k) is a symmetric matrix. Proof. In fact, BT1 UL1 ¼ L2 UBT1 , hence, one has " # " # 0 B1 L2 0 L1 B1 BUL ¼ ¼ ¼ LUB, BT1 L1 0 L2 BT1 0
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namely, B and L are commutative matrices, which implies that B and eLt are commutative T matrices. According to (A4) and the fact that ðeLt ÞT ¼ eL t , it follows that T Z hkþ1 C T ðkÞ ¼ eLðhkþ1 hk Þ þ eLðhkþ1 tÞ B dt hk
T
Z
hkþ1
T
þ BT eL ðhkþ1 tÞ dt hk Z hkþ1 BeLðhkþ1 tÞ dt ¼ eLðhkþ1 hk Þ þ hk Z hkþ1 eLðhkþ1 tÞ B dt ¼ CðkÞ, ¼ eLðhkþ1 hk Þ þ ¼ eL
ðhkþ1 hk Þ
ð18Þ
hk 0
which shows that C (k) is a symmetric matrix. & Remark 3. The condition L1UB1 ¼ B1UL2 is called a joining condition between two different groups. On this condition, B and L are commutative matrices, which reflects equivalent relation between two different groups. Furthermore, the following results are based on the joining condition. 0
Considering C(k), C (k), one obtains that " # " #T U1 0 U1 0 Lðhkþ1 hk Þ U ¼ diagf1,. . .,eln ðhkþ1 hk Þ , 1,. . .,elnþm ðhkþ1 hk Þ g, Ue 0 U2 0 U2 ð19Þ "
U1
0
0
U2 Z
# Z U
hkþ1
¼ hkþ1
¼ hk
U1 0
hk
Z
e
hk
"
"
"
hkþ1
U1 0
Lðhkþ1 tÞ
B dtU
U1
0
#T
U2 " #T 0 U1 0 Lðhkþ1 tÞ dtUB Ue U2 0 U2 # " #T " U1 0 U1 0 Lðhkþ1 tÞ U dtU Ue U2 0 U2 0 0
#
# " 0 U1 UBU U2 0 lnþm ðhkþ1 hk Þ
1eln ðhkþ1 hk Þ 1e ¼ diag hkþ1 hk ,. . ., , hkþ1 hk ,. . ., ln lnþm " # T 0 U1 B 1 U2 : U2 BT1 U1T 0 Therefore, according to (A3), one has " T # " u11 0 T U1 B1 U2 ¼ T UB1 U½u21 ,u22 ¼ u12 0 where P1 ¼ uT12 B1 u22 2 Rðn1Þðm1Þ :
0 P1
0 U2
#T
ð20Þ
# ,
ð21Þ
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Denote
C1 ðkÞ ¼ diag el2 ðhkþ1 hk Þ ,. . .,eln ðhkþ1 hk Þ ,
l ðh h Þ C2 ðkÞ ¼ diag e nþ2 kþ1 k ,. . .,elnþm ðhkþ1 hk Þ , 1el2 ðhkþ1 hk Þ 1eln ðhkþ1 hk Þ ,. . ., , D1 ðkÞ ¼ diag l2 ln 1elnþ2 ðhkþ1 hk Þ 1elnþm ðhkþ1 hk Þ ,. . ., : D2 ðkÞ ¼ diag lnþ2 lnþm 0
Due to the symmetry of C (k), one has D2 ðkÞPT1 ¼ PT1 D1 ðkÞ, and then system (17) could be rewritten as 2 3 1 0 0 0 6 7 C1 ðkÞ 0 D1 ðkÞP1 7 60 6 7yðkÞ: ð22Þ yðk þ 1Þ ¼ 6 7 0 1 0 40 5 0 PT1 D1 ðkÞ 0 C2 ðkÞ The following theorem gives a necessary and sufficient condition for the couple-group average-consensus. Theorem 1. System (6) solves the couple-group average-consensus problem if and only if " C1 ðtÞ jjW ðtÞjj ¼ T P1 D1 ðtÞ
D1 ðtÞP1 C2 ðtÞ
# o1,
8t 2 ½t0 ,t1 ,
ð23Þ
where
C1 ðtÞ ¼ diag el2 t ,. . .,eln t , C2 ðtÞ ¼ diag elnþ2 t ,. . .,elnþm t , 1el2 t 1eln t ,. . ., : D1 ðtÞ ¼ diag l2 ln
Proof. To prove the sufficiency, we first prove that system (9) solves the couple-group average-consensus problem. Let 2 3 1 0 0 0 60 0 1 0 7 6 7 Zðk þ 1Þ ¼ 6 7yðk þ 1Þ, 0 5 4 0 In1 0 0 0 0 Im1 then
2
1
0
0
6 60 Zðk þ 1Þ ¼ 6 60 4 0
1 0
0 C1 ðkÞ
0
PT1 D1 ðkÞ
0
3
2 1 7 0 7 7zðkÞ ¼ 6 4 D1 ðkÞP1 7 5 C2 ðkÞ
3 1 W ðkÞ
7 5zðkÞ
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2
3
1
6 6 ¼6 6 4
1 k Y
7 7 7Zð0Þ, 7 W ðiÞ 5
ð24Þ
i¼0
Hence, 2
2
3
Z1 ðk þ 1Þ 6 6 6 Z ðk þ 1Þ 7 lim 4 2 5 ¼ lim 6 k-1 k-16 4 Z3 ðk þ 1Þ
3 Z1 ð0Þ 7 7 Z1 ð0Þ 6 Z ð0Þ 7 76 6 (2 7 ) 7: 7 6 Z ð0Þ ¼ lim 2 k 4 5 k
Y Y 7 7 6 k-14 W ðiÞ 5 Z ð0Þ W ðiÞ Z3 ð0 5 3
1 1
2
2
3
3
i¼0
i¼0
ð25Þ Due to the continuity of :W(t):, there exists t0A[t0,t1], such that jjW ðt0 Þjj ¼ maxt2½t0 ,t1 jjW ðtÞjj. Denote M0 ¼ :G(t0:, and thus it follows from the condition (23) that M0o1. Therefore, ( ) Y Y k k lim jjZ3 ðk þ 1Þjj ¼ lim W ðiÞ Z3 ð0Þr lim W ðiÞUjjZ3 ð0Þjj k-1 k-1 k-1 i¼0 i¼0 r lim M0kþ1 UjjZ3 ð0Þjj ¼ 0:
ð26Þ
k-1
Namely, limk-1 Z3 ðk þ 1Þ ¼ 0. According to (25), one has 3 2 3 2 3 2 Z1 ðk þ 1Þ y1 ð0Þ Z1 ð0Þ 7 6 7 6 7 6 lim 4 Z2 ðk þ 1Þ 5 ¼ 4 Z2 ð0Þ 5 ¼ 4 y3 ð0Þ 5, k-1 Z3 ðk þ 1Þ 0 0 2
ð27Þ
1
0
0
0
32
60 6 lim yðk þ 1Þ ¼ 6 40 k-1 0
0 1
In1 0
0 0
76 y3 ð0Þ 7 6 07 76 7 6 7 76 7¼6 7, 54 0 5 4 y3 ð0Þ 5
0
0
Im1
" lim xðk þ 1Þ ¼
k-1
U1
0
0
U2
#T
2
3 y1 ð0Þ 6 0 7 6 7 U6 7 4 y3 ð0Þ 5 0
y1 ð0Þ
0
3
2
y1 ð0Þ
0
3 ð28Þ
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! 3 n 1 X 3 6 2 xi ð0Þ 7 1 7 6 n i¼1 pffiffiffi y1 ð0Þ 7 6 7 6 6 n 7 7 6 6 ^ 7 7 6 6 ! 7 ^ 7 6 6 n 7 7 6 1 X 6 1 7 6 pffiffiffi y ð0Þ 7 6 x ð0Þ 7 i 7 6 1 n 7 6 7 6 6 n i¼1 7 6 !7 ¼ ¼6 1 7: 6 7 nþm X 7 1 6 pffiffiffiffi y3 ð0Þ 7 6 6 xi ð0Þ 7 7 6 6 m 7 7 6m 6 7 i ¼ nþ1 7 6 6 7 ^ 7 6 6 7 ^ 7 6 6 5 6 4 1 !7 7 nþm pffiffiffiffi y3 ð0Þ X 7 61 m 4 xi ð0Þ 5 m i ¼ nþ1
587
2
ð29Þ
Thus, it is proved that system (9) solves the couple-group average-consensus problem. Therefore, according to Lemma 2, system (6) solves the couple-group average-consensus problem. Next, we apply the reduction to absurdity to prove the necessity. Suppose that Eq. (23) is 0 not true, then there exists t A[t0,t1], such that :W ðt0 Þ:Z1. Therefore, there also exists an 0 eigenvector Z , such that :Z 0 : ¼ 1 and :W ðt0 Þz0 :Z1. From Eq. (26), if one chooses 0 0 Z3(0) ¼ Z and hkþ1hk ¼ t for all k, then Z3(kþ1) could not converge to zero as k-N. As a result, system (9) could not solve the couple-group average-consensus problem. According to Lemma 2, system (6) also could not solve the couple-group averageconsensus problem, and this results in a contradiction. This completes the proof. & In Theorem 1, the condition (23) is a necessary and sufficient condition. However, it is difficult to be implemented in practical applications, because we should verify that every point in [t0,t1] satisfies the condition (23). On the other hand, since the matrices L1, L2 are given in the practical situation, we should design appropriate parameters t0, t1, and matrix B such that system (6) solves the couple-group average-consensus problem. In view of these points, we give the algebraic criteria about the couple-group average-consensus problem in Theorem 2.
Theorem 2. Given system (6), if rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :B:F 1 n þ m2 , o 2 n þ m3 l2 then there exists T0 ¼ T0(n, m, L1, L2), such that for all t0A[T0,N), system (6) solves the couple-group average-consensus problem. Proof. Consider the auxiliary function rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 jtrW ðtÞj2 , gðtÞ ¼ jjC1 ðtÞjj2F þ jjC2 ðtÞjj2F þ 2jjD1 ðtÞP1 jj2F n þ m2
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Then n þ m3 2 jtrW ðtÞj2 1 jtrW ðtÞj2 2 g ðtÞ ð n þ m3 Þg ¼ ð t Þ n þ m2 n þ m2 n þ m2 ðn þ m2Þ2 1 ðn þ m3Þ jjC1 ðtÞjj2F þ jjC2 ðtÞjj2F þ 2jjD1 ðtÞP1 jj2F jtrW ðtÞj2 n þ m2 h 2 1 ðn þ m3Þ jjC1 ðtÞjj2F þ jjC2 ðtÞjj2F trW ðtÞ ¼ n þ m2 þ2ðn þ m3ÞjjD1 ðtÞP1 jj2F ,
¼
ð30Þ
Because of jtrW ðtÞj2 rðn þ m2Þ½jjC1 ðtÞjj2F þ jjC2 ðtÞjj2F , one has n þ m3 2 jtrW ðtÞj2 1 g ðtÞ 2ðn þ m3ÞjjD1 ðtÞP1 jj2F Z 2 n þ m2 n þ m2 ðn þ m2Þ jjC1 ðtÞjj2F jjC2 ðtÞjj2F :
ð31Þ
Moreover, 1 2ðn þ m3ÞjjD1 ðtÞP1 jj2F jjC1 ðtÞjj2F jjC2 ðtÞjj2F t-þ1 n þ m2 2ðn þ m3Þ jjD2 ð1ÞP1 jj2F 40: ¼ n þ m2 lim
ð32Þ
According to Eqs. (31) and (32), there exists T00 ¼ T00 ðn,m,L1 ,L2 Þ such that for all t 2 ½T00 ,1Þ n þ m3 2 jtrW ðtÞj2 g ðtÞ4 : n þ m2 ðn þ m2Þ2
ð33Þ
By the property of Frobenius norm, it is easy to obtain that " # 0 0 2 1 jjP1 jj2F ¼ ¼ jjU1 B1 U2T jj2F ¼ jjB1 jj2F ¼ jjBjj2F , 0 P1 2
ð34Þ
F
lim g2 ðtÞ ¼ lim jjC1 ðtÞjj2F þ jjC2 ðtÞjj2F þ 2jjD1 ðtÞP1 jj2F
t-þ1
t-þ1
¼ 2jjD2 ð1ÞP1 jj2F r
2jjP1 jj2F jjBjj2F ¼ : l22 l22
From Eq. (35) and the condition rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :B:F 1 n þ m2 , o 2 n þ m3 l2
1 jtrW ðtÞj2 n þ m2
ð35Þ
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589
there exists T 00 0 ¼ T 00 0 ðn,m,L1 ,L2 Þ such that for all t 2 ½T 00 0 ,1Þ the following inequality hold n þ m3 2 1 g ðtÞo : n þ m2 4
ð36Þ
Let T0 ¼ maxfT00 ,T 00 0 g, for all tA[T0,N), one has rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi trW ðtÞ n þ m3 gðtÞo1: þ n þ m2 n þ m2
ð37Þ
Now, it follows from Theorem 1 in [33] that for t0A[T0,N), the eigenvalues of G(t) belong to (1,1), namely " # C1 ðtÞ D1 ðtÞP1 :W ðtÞ: ¼ T o1, 8t 2 ½t0 ,t1 : C2 ðtÞ P1 D1 ðtÞ By applying Theorem 1, system (6) solves the couple-group average-consensus problem. This completes the proof. & Remark 4. In Theorem 2, the algebraic criterion rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :B:F 1 n þ m2 o 2 n þ m3 l2 is simple to implement, and the parameter T0 is related to the communication topology and the number of agents in different groups. If we consider the information exchanges from external groups as disturbances, it can be inferred from Theorem 2 that the couplegroup average-consensus of multi-agents system can still be solved when the disturbances are sufficiently small and the intervals of the update time are adequately long. Due to the sufficiency of the criteria, there may exist certain extent of conservatism in the selection of T0. Obviously, we have T0a0, in fact, if T0 ¼ 0, W(0) is an identity matrix, namely, :W ðT0 Þ: ¼ 1, which does not satisfy condition (23). In the next section, it can be found that T0 is close to 0, which indicates that less conservative results can be obtained. The following corollary of Theorem 1 gives the convergence rate of hybrid protocol in group consensus problem. Corollary 3. Given system (9), jjxðk þ 1Þxn jjrM1kþ1 Ujjxð0Þjj, where 1elt0 1elt1 lt0 lt1 jjBjj, e jjBjj , þ þ M1 ¼ max e l l with l ¼ minfl2 ,lnþ2 g. Furthermore, if 8k 2 N þ ,
hkþ1 hk ¼ T0 , then
kþ1 1elT0 jjBjj Ujjxð0Þjj: jjxðk þ 1Þxn jjr elT0 þ l
ð38Þ
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Proof. Since " yðkÞ ¼
U1
0
0
U2
"
U1 y ¼ 0 n
2
# xðkÞ,
1
0
60 0 6 Zðkþ1Þ ¼ 6 4 0 In1 0 0
0
0
3
1 0 7 7 7yðkþ1Þ, 0 0 5 0 Im1
# 0 xn U2
and 2
1
0
60 0 6 Zn ¼ 6 4 0 In1 0 0
0
0
3
1 0
0 0
7 7 n 7y , 5
0
Im1
one has jjxðk þ 1Þxn jj ¼ jjyðk þ 1Þyn jj ¼ jjZðk þ 1ÞZ n jj 2 3 1 2 3 1 6 7 1 6 7 6 7 6 7 1 ¼ 6 k 5Zð0Þ Y 7Zð0Þ4 4 W ðiÞ 5 0 i¼0 2 3 0 " # 6 7 k 0 Y 6 7 7UjjZð0Þjjr r6 rðW ðiÞÞ jjxð0Þjj, k Y 6 7 4 5 i¼0 W ðiÞ i¼0 Then considering r(W(t)) with tA[t0,t1], one obtains " C1 ðtÞ rðW ðtÞÞ ¼ T P1 D1 ðtÞ
D1 ðtÞP1 C2 ðtÞ
#
2 3 2 3 0 0 # " # 6 7 6 7 " 0 B 1 C1 ðtÞ D1 ðtÞ 6 7 6 7 U1 7þ6 7U ¼ 6 U T 7 7 6 6 B1 0 0 0 U2 4 5 4 5 C2 ðtÞ D2 ðtÞ 2 3 2 3 0 0 # 6 7 6 7 " D1 ðtÞ C1 ðtÞ 6 7 6 7 0 B1 7 7 6 6 r 6 7 þ 6 7U BT 0 0 0 1 4 5 4 5 C2 ðtÞ D2 ðtÞ
ð39Þ
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" # " # " C1 ðtÞ D1 ðtÞ 0 ¼ þ U C2 ðtÞ D2 ðtÞ BT1 Let l ¼ minfl2 ,lnþ2 g, and " # C1 ðtÞ lt re , C2 ðtÞ
# B1 : 0
591
ð40Þ
" # D1 ðtÞ rð1elt =lÞ, D2 ðtÞ
then rðW ðtÞÞrelt þ
1elt jjBjj: l
ð41Þ
0
Denote f(t)¼ eltþ((1elt)/l)99B99, it follows f (t)¼ [99B99l] elt, hence, f(t) is a monotonic function, let M1 ¼ maxt0 rtrt1 f ðtÞ ¼ maxff ðt0 Þ,f ðt1 Þg, then r(W(t))rM1. According to Eq. (39), one has jjxðk þ 1Þxn jjrM1kþ1 Ujjxð0Þjj,
ð42Þ
Moreover, if 8k 2 N þ , hkþ1hk ¼ T0, then M1 equals to elT0 þ ð1elT0 =lÞ:B:. This completes the proof. & Remark 5. According to Corollary 1, the convergence rate of group consensus depends on the parameters l, t0, t1 and B. Therefore, this property can be improved by selecting proper values of these parameters. Obviously, the result in [10] is a special case of the result (38). In fact, if :B: ¼ 0, the information exchange between different groups does not exist, Eq. (38) is then rewritten as jjxðk þ 1Þxn jjrelðkþ1ÞT0 Ujjxð0Þjj, which is consistent with the result in [10].
Furthermore, the couple-group average-consensus problem can be extended to a more general case, i.e. the agents in a network reach more than two consistent states asymptotically. Without loss of generality, suppose that a complex network (G,x) is composed of m subnetworks, and the qth sub-network has nq agents with a corresponding topology graph Gq, q ¼ 1,2,. . .,m, then a multi-group consensus can be defined as limt-1 jjxi xj jj ¼ 0, 8i, jAGq, q ¼ 1,2,. . .,m. For the complex network (G,x), assume that the topology graph G is undirected, and then a new hybrid protocol can be defined as follows: X X ui ðtÞ ¼ bij xj ðhk Þ þ þ aij ðxj xi Þ þ vj 2N1i
þ
X
vj 2Nqi
bij xj ðhk Þ,
8i 2 Gq ,
t 2 ½hk ,hkþ1 Þ,
k 2 Nþ
vj 2Nmi
where the sequence{hk} satisfies the assumptions (A1) and (A2).
ð43Þ
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Given the hybrid protocol (43), the multi-agent system (1) evolves according to the following system: X X bij xj ðhk Þ þ þ aij ðxj xi Þ þ x_ i ðtÞ ¼ vj 2 N1i
þ
vj 2 Nqi
X
bij xj ðhk Þ,
8i 2 Gq ,
t 2 ½hk ,hkþ1 Þ,
k 2 N þ,
ð44Þ
vj 2 Nmi
Then Eq. (44) can be translated into a matrix form _ ¼ LxðtÞ þ Bxðhk Þ, xðtÞ where
2
6 6 L¼6 6 4
8t 2 ½hk ,hkþ1 Þ,
k 2 N þ,
ð45Þ
3
L1 L2 & Lp
7 7 7, 7 5
with Lq 2 Rnq nq being the Laplacian matrix of sub-network Gq, and the matrix 3 2 0 B12 B1m 7 6 T 0 B2m 7 6 B12 7 B¼6 6 ^ ^ & ^ 7 5 4 BT1m
BT2m
0
describes the information exchanges among different groups. For instance, Bqi 2 Rnq ni represents the information exchanges between the qth group and the ith group. Here Bqi also satisfies the assumption (A3), i.e. 1T UBqi ¼ 0,
Bqi U1 ¼ 0:
ð46Þ
The following two results are generalizations of Lemmas 2 and 3. Lemma 4. System (45) solves the multi-group average-consensus problem if and only if system (9) solves the multi-group average-consensus problem. 0
Lemma 5. Given system (17), if LqUBqi ¼ BqiULi, 8q, i, then C (k) is a symmetric matrix, where 2 3 2 3T U1 U1 6 7 6 7 U2 U2 6 7 6 7 C 0 ðkÞ ¼ 6 7UCðkÞU6 7 , & & 4 5 4 5 Um Um 3 2 0 7 6 lðqÞ 7 6 2 7 Uq ULq UUqT ¼ 6 7 6 & 5 4 ðqÞ ln q ðqÞ ðqÞ and 0olðqÞ 2 rl3 r rlnq are eigenvalues of Lq.
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Remark 6. The condition Lq Bqi ¼ Bqi Li, 8q, i is called a joining condition between different groups, which means that different groups are equivalent. Under this condition, both B and L are commutative matrices. Furthermore, based on the joining condition, the following results are obtained. pffiffiffiffiffi pffiffiffiffiffi Obviously, the first column vectors of UqT is uq1 ¼ ½ð1= nq Þ,. . .,ð1= nq ÞT . According to Eq. (44), we have " T # " # uq1 0 0 T Uq Bqi Ui ¼ T UBqi U½ui1 ,ui2 ¼ , 0 Pqi uq2 where Pqi ¼ uTq2 Bqi ui2 2 Rðnq 1Þðni 1Þ . The following result is a generation of Theorem 1. Theorem 3. System (45) solves the multi-group average-consensus problem if and only if 2 C1 ðtÞ D1 ðtÞP12 6 T C2 ðtÞ 6 P12 D1 ðtÞ jjW ðtÞjj ¼ 6 6 ^ ^ 4 T P1m D1 ðtÞ PT2m D2 ðtÞ
3 D1 ðtÞP1m 7 D2 ðtÞP2m 7 7o1, 7 & ^ 5 Cm ðtÞ
8t 2 ½t0 ,t1 ,
ð47Þ
where n ðqÞ o ðqÞ Cq ðtÞ ¼ diag el2 t ,. . .,elnq t ,
( Dq ðtÞ ¼ diag
ðqÞ
1el2 lðqÞ 2
ðqÞ
t
,. . .,
1elnq t lðqÞ nq
) ,
8q 2 f1,2,. . .,mg: In Theorem 3, the condition (47) is a necessary and sufficient condition. However, it is difficult to implement because we should verify that every point in [t0,t1] satisfies the condition (47). Therefore, inspired by the above analysis, the following theorem gives the algebraic criteria about the multi-group average-consensus problem, and the proof of which is similar to Theorem 2. Theorem 4. Given system (45), if !ð1=2Þ Pm jjBjjF 1 q ¼ 1 nq m Pm o , 2 l q ¼ 1 nq m1 ðmÞ where l ¼ minflð1Þ 2 ,. . .,l2 g, there exists T0 ¼ T0 ðn1 ,. . .,nm ,L1 ,. . .,Lm Þ, such that for all t0A[T0,N), system (45) solves the multi-group average-consensus problem.
4. Simulation results In this section, an example is provided to illustrate the effectiveness of the proposed theoretical results. We consider five agents with the interaction topology given in Fig. 2.
594
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1
1/3 G1
7/8 -7/8
1/3 -7/16 7/16
2
4
1/2
-7/16
G2
5 3
7/16
Fig. 2. The interaction topology.
From Fig. 2, the Laplacian matrixes of G1 and G2 can be easily obtained as follows: 3 2 2 1 1 2 3 6 3 1 1 3 37 7 6 7 7 6 1 1 6 27 7, L2 ¼ 6 2 0 L1 ¼ 6 7 6 3 3 4 1 1 5 7 6 4 1 1 5 2 2 0 3 3 and the eigenvalues of L1 and L2 are l1 ¼ 0, l3 ¼ 1, l4 ¼ 0, l5 ¼ 1. The information exchange between G1 and G2 is 3 2 7 7 6 8 87 7 6 6 7 7 7 7 6 B1 ¼ 6 7 6 16 16 7 4 7 7 5 16 16 According to Eqs. (31) and (34) in Theorem 2, it can be concluded that system (6) solves the couple-group average-consensus problem for all t0A[0.2959,N). We choose t0 ¼ 0.3, t1 ¼ 0.4, and the stochastic sampling intervals are shown in Fig. 3. With the initial T state xð0Þ ¼ 0:1 0:2 0:3 0:4 0:5 ,the couple-group average-consensus is reached asymptotically, and the state trajectories of five individual agents are shown in Fig. 4. 5. Conclusion A group consensus problem with discontinuous information transmission between different groups of dynamic agents was investigated. In the presence of undirected information exchange, this paper provided sufficient and necessary conditions for solving the couple-group average-consensus problem asymptotically by using the hybrid protocol. Moreover, by employing the tools of algebraic graph theory, matrix theory, and control theory, we performed the convergence analysis and established several algebraic criterions for the group consensus problem. In addition, as an extension of the couple-group consensus problem, the multi-group consensus problem was introduced and the corresponding analysis
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595
0.42
0.4
Sampling Intervals
0.38
0.36
0.34
0.32
0.3
0
5
10
15
20
25
30
Time
Fig. 3. The stochastic sampling intervals.
agent 1 agent 2 agent 3 agent 4 agent 5
0.6
States of agents
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
30
Time
Fig. 4. State trajectories of five individual agents.
results were derived by the similar techniques. Finally, the simulation confirmed the theoretical results. Further extensions will focus on disturbance rejection and robustness properties of the hybrid protocol, as well as switching network topologies between agents, communication dropouts and message transmission delay, and so on. Acknowledgment This work was supported by the National Natural Science Foundation of China under the Grants nos. 60974017, 61273212, and 61004097.
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[27] J. Yu, L. Wang, Group consensus of multi-agent systems with switching topologies and communication delays, Systems & Control letters 59 (6) (2010) 340–348. [28] C. Godsil, G. Royle, Algebraic Graph Theory, Springer-Verlag, New York, 2001. [29] L. Huang, Linear Algebra in Systems and Control Theory, Science Press, Beijing, 1984. [30] W. Ren, R.W. Beard, Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Transaction on Automatic Control 50 (5) (2005) 655–661. [31] J. Cortes, Discontinuous dynamical systems, IEEE Control Systems 28 (3) (2008) 36–73. [32] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, 1982. [33] Y. Gu, The distribution of eigenvalues of a matrix, Acta Mathematicae Applicatae Sinica 17 (4) (1994) 501–511.
Journal of the Franklin Institute 350 (2013) 575–597 www.elsevier.com/locate/jfranklin
Group consensus in multi-agent systems with hybrid protocol Hong-xiang Hu, Li Yun, Wen-An Zhang, Haiyu Song Department of Automation, Zhejiang University of Technology, Zhejiang Provincial United Key Laboratory of Embedded Systems, Hangzhou 310023, PR China Received 26 February 2011; received in revised form 19 September 2012; accepted 13 December 2012 Available online 15 January 2013
Abstract This paper investigates a group consensus problem with discontinuous information transmissions among different groups of dynamic agents. In the group consensus problem, the agents reach more than one consistent state asymptotically. We consider that the communication topology of these agents, represented by a network, is undirected. Then a novel group consensus protocol, called hybrid protocol, is proposed to solve the couple-group average-consensus problem. The convergence analysis is presented and the algebraic criterions are established. Furthermore, the multi-group consensus is discussed as an extension of the couple-group consensus. By similar techniques, some analysis results are presented. The analysis tools developed in this paper are based on algebraic graph theory, matrix theory, and control theory. Finally, the simulations are provided to demonstrate the effectiveness of the proposed theoretical results. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Recently, distributed coordination of dynamical multi-agent systems has attracted multidisciplinary researchers in various areas including system control theory, statistical physics, biology, applied mathematics, and computer science. One of the most important issue is the agreement or consensus of multi-agent systems, which has been widely applied in many areas such as cooperative control of unmanned air vehicles, formation control, flocking of social insects, wireless sensor networks, mobile robotic swarms, etc. [1–7]. n
Corresponding author. E-mail address: [email protected] (L. Yu).
0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.12.020
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Generally, the main objective of consensus problem is to design an appropriate algorithm or interaction rule such that a group of agents converges to a consistent quantity of interest. The algorithm or the interaction rule is usually called agreement protocol. The consistent quantity that depends on the initial states of all agents is called consensus state which might represent physical quantities such as altitude, position, temperature, voltage, etc. Undoubtedly, the consensus state and the convergence rate are crucial for the consensus problem, which indicate where the multiple dynamic agents reach and how quickly they achieve it, respectively. There have been lots of works studying the consensus problem in the literature. For example, Vicsek et al. [8] presented a discrete-time model containing finite autonomous agents that move in the plane with same speed but different headings, where the concept of neighbors of agents was introduced and some simulations were provided to demonstrate the efficiency of the nearest neighbor rule. Jadbabaie et al. [9] provided a formal analysis for the emergence of alignment in a simplified Viscek’s model of flocking. Olfati-Saber and Murray [10] addressed a systematical framework for consensus problem of dynamic agents with fixed/switching network topology and communication time-delays. In their work, the second smallest eigenvalue of the Laplacian matrix is first used to quantify the convergent speed of the consensus algorithm. More recently, Xiao and Wang [11] studied such asynchronous consensus problem in a continuous-time multi-agent system with discontinuous information transmission by using nonnegative matrix theory and graph theory. Rahmani et al. [12] investigated the controlled agreement problem for multi-agent networks, where several sufficient graphical conditions were obtained based on the graph automorphisms for the system’s uncontrollability. Based on the work introduced in [13,14], Galvan-Guerra et al. [15] studied the specific optimal control problem associated with a multiagent dynamic system. In their work, to minimize the tracking error in the leader-follower model, the hybrid LQ-based optimization techniques were considered as an auxiliary method associated with the constructive solution procedure for the initial multiagent tracking problem. All these results are only concerned with first-order dynamics. Since many practical individual systems, especially the mechanical systems, are of secondorder dynamics, there have been also some related works focusing on the consensus problem of the agents with this kind of dynamics [16–18]. Meanwhile, a variety of methods have been presented in the literature to analyze the convergence rate of the consensus of multi-agent systems. Generally, it is considered that communication topology plays a key role in the convergence of consensus process. For example, Cao et al. presented new graph-theoretic results appropriate for the analysis of a variety of consensus problems cast in dynamically changing environments in [19,20], where the concepts of rooted graph, strongly rooted graph, and neighbor-shared graph were introduced and the worst convergence rates of these graphs were derived. Nedic et al. [21] considered a constrained consensus problem and proposed a distributed projected consensus algorithm with the results of convergence rate being established. Since in some practical situations it may be required that the agreement is reached in finite time, finite-time agreement becomes an important topic over the past few years [22–25]. However, almost all the aforementioned results were only concerned with one common consensus state, which need to be further generalized in several cases. For example, in the formation flight of Unmanned Air Vehicles (UAV), the set of UAV must be divided into sub-sets when some cooperative tasks need to be accomplished synchronously. Therefore, appropriate protocols and algorithms need to be designed such that agents in a network
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reach more than one common consensus state. Fortunately, this problem has been already addressed by Yu and Wang [26,27], which is named as the group consensus problem. In the group consensus problem, the whole network is divided into multiple sub-networks, and all the agents are divided into multiple groups consequently. Moreover, information exchange exists in different groups. Furthermore, it was assumed in [27] that the channels between different groups must exist continuously. Thus, the protocols proposed in [27] are pure continuous-time ones. In practice, the information exchange between different groups may be intermittent due to unreliable communication channels. To the best of our knowledge, the group consensus problem of multi-agent systems with intermittent information exchanges has not yet been investigated in the existing literature, which motivates the present research. In contrast to the protocol presented in [27], this paper investigates a group consensus problem with hybrid protocol. The hybrid protocol includes continuous-time signal that depicts the information exchange in the same group, and discrete-time signal that describes the information exchange between different groups, which is considered more consistent with practical situations. An outline of this paper is as follows. Some basic definitions and results in matrix and graph theories are presented in Section 2. The main results about group consensus and convergence rate are given in Section 3. The results are verified via numerical simulations in Section 4. Finally, the paper is concluded in Section 5. Notations. The notation used in this paper is fairly standard. Specifically, Denote by R and N the sets of real numbers and natural numbers, respectively. Denote by Rn the set of n 1 real column vectors and ( )T the transpose. Meanwhile, N þ ¼ f0g [ N,
1n ¼ ð1,1, 1 ,0 0ÞT 2 Rnþm , |fflfflfflfflffl{zfflfflfflfflffl} n
1m ¼ ð0, 0,1,1, 1 ÞT 2 Rnþm : |fflfflfflfflffl{zfflfflfflfflffl} m
Denote by InARn n the identity matrix and : : the Euclidean vector norm. For AARn n, denote by tr(A), :A:, and r(A) the trace, the norm, and the spectral radius of A, respectively. Specially, denote by :A:F the Frobenius norm. Then we have jjAjj ¼ sup x 2 Rn jjAxjj, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 jjAjjF ¼ i,j ¼ 1 jaij j , and rðAÞ ¼ maxfjli jg with li being the eigenvalue of A.
jjxjj ¼ 1
i
2. Problem formulation Some basic concepts of graph theory and results in matrix theory, which have been used in [28,29], are introduced in this section. In a multi-agent system, each agent can communicate with several other agents which are defined as its neighbors. The topology of such communication relationships then can be represented by a weighted undirected graph G ¼ (V, x, A) of order n with a set of nodes V ¼ fv1 ,v2 ,. . .,vn g, a set of edges xDV V, and a nonnegative symmetric matrix A ¼ [aij]. The node indexes belong to a finite index set I ¼ f1,2,. . .,ng. And the edge of G is denoted by eij ¼ (vi,vj), which is an unordered pair of vertices. (vi,vj)Ax3aij403 agent i and j can communicate with each other, namely, they are adjacent. Moreover, we assume that aii ¼ 0
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for all iAI. Here, A is called the weight matrix and aij is the weight of edge eij ¼ (vi,vj). The set of neighbors of node vi is denoted by Ni ¼ {vj9(vj,vi)Ax}. A path in a graph from vi to vj is a sequence of distinct vertices from vi to vj such that the consecutive vertices are adjacent. A graph is connected if there is at least one path between any two vertices in the graph. Let xiAR denote the state of node vi, we refer to (G,x) with state x ¼ ½x1 ,x2 ,. . .,xn T 2 Rn and topology graph G. Each agent updates its current state based upon the information of its neighbors. Suppose that each agent of a graph has the dynamics as follows: x_ i ðtÞ ¼ ui ðtÞ,
8i 2 I,
ð1Þ
where ui ðtÞ ¼ fi ðxi ,xj1 ,. . .,xjm Þ is a state feedback. If the cluster J ¼ fxj1 ,. . .,xjm gDNi , then ui(t) is said to be a protocol with topology G. Ren and Beard [30] adopted the following protocol to solve the consensus problem and the average consensus problem: X aij ðxj xi Þ, ð2Þ ui ðtÞ ¼ vj 2Ni
It has been proved that if the topology G is undirected and connected, then the protocol (2) asymptotically solves an average consensus problem [10]. Considering practical applications of cooperative control, a more general issue, i.e., the group consensus, is addressed in this paper, where agents in a network reach more than one common consensus state asymptotically. Without loss of generality, we consider the case that agents in a network reach two consistent states asymptotically. Let a complex network (G,x) consists of nþm (n,m41) agents, the group consensus problem is simply defined as follow: the first n agents reach a consistent state while the other m agents reach another consistent state in the presence of information exchange between the two groups. Denote I1 ¼ f1,2,. . .,ng, I2 ¼ fn þ 1,n þ 2,. . .,n þ mg, V1 ¼ fv1 ,v2 ,. . .,vn g, V2 ¼ fvnþ1 ,. . .,vnþm g, N1i ¼ {vjAV19(vj,vi)Ax}, and N2i ¼ {vjAV29(vj,vi)Ax}. Obviously, I¼ I1[I2, V¼ V1[V2, and Ni ¼ N1i[N2i. Then define a new protocol for a complex network (G,x) as follows: P P ( 8i 2 I1 , vj 2N1i aij ðxj xi Þ þ vj 2N2i bij xj ðhk Þ P P ui ðtÞ ¼ , 8t 2 ½hk ,hkþ1 Þ, k 2 N þ , 8i 2 I2 , vj 2N1i bij xj ðhk Þ þ vj 2N2i aij ðxj xi Þ ð3Þ where aijZ0, bijAR, and hk is the update time, which is the time when the information of neighbors from different groups known by agent i is updated. In other words, the update time is the moment when information exchange between different groups takes place. Assume that {hk} satisfies the following conditions: (A1) there exist positive real numbers t0, t1 such that for any kAN, t0rhkþ1hkrt1; (A2) h0 ¼ 0. Since the protocol (3) contains both continuous-time and discrete-time signals, it is called a hybrid protocol. Moreover, the assumptions (A1) and (A2) in [27] are also satisfied
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between different groups, namely (A3) nþm X j ¼ nþ1
bij ¼ 0,
n X
bij ¼ 0:
j¼1
Remark 1. The structure of different groups is shown in Fig. 1, which shows that the protocol (3) can be implemented in the following way: when t¼ hk, the switch is closed, and the information exchange between different groups takes place; when tA(hk,hkþ1), the switch is open, and the information exchange between different groups maintains the value at t ¼ hk due to the zero-order hold. This protocol is different from that introduced in [27], which is a pure continuous-time protocol, that is, channels between different groups must exist continuously. Since in reality the channels between different groups may be unreliable due to disturbances, especially when one group is far away from another, and thus may result in discontinuous information exchanges, the proposed hybrid protocol (3) is more suitable for practical applications than that in [27]. Remark 2. The protocol (3) indicates that an agent may receive the information of agents in another group at the time hk. Meanwhile, one has hk ¼ t(thk), and the term thk could be considered as time-delay, hence the protocol (3), in essence, is a time-delay protocol. However, it is easy to see that different from Eq. (18) in [27], this time-delay is not a continuous function of t, but only a piecewise continuous function. Therefore, the solution of systems (1) and (3) is referred to as sample-and-hold solution [31]. In order to obtain the main results, the following definitions are needed. Definition 1. (Yu and Wang [26]). The hybrid protocol (3) is said to asymptotically solve a couple-group consensus problem, if there exists an x * Aspan{1n, 1m}, such that limt-1 jjxðtÞxn jj ¼ 0, where xðtÞ ¼ ðx1 ,x2 ,.. .,xnþm be the initial xi(0) P PÞ: Furthermore, let state of agent i, if x * ¼ a1nþb1m, where a ¼ 1=nÞ ni ¼ 1 xi ð0Þ and b ¼ 1=mÞ nþm i ¼ nþ1 xi ð0Þ, then it is said that the hybrid protocol (3) asymptotically solves a couple-group averageconsensus problem. Definition 2. (Yu and Wang [27]). A network with topology G1 ¼ (V1, x1, A1) is said to be a sub-network of a network with topology G ¼ (V, x, A) if (i) V1DV, (ii) x1Dx and (iii) the weighted adjacency matrix A1 inherits A. Furthermore, if the inclusion relations in (i) and (ii) are strict, then it is said that G1 is a proper sub-network of G.
ZOH Group 1
Switch ZOH Network
Fig. 1. Multi-agent systems with hybrid protocol.
Group 2
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According to Definition 2, the first n agents constitute a network G1, and the other m agents constitute another network G2. It is assumed that (A4) network G1 and G2 are connected. Before presenting the main results, a useful lemma is introduced as follows. Lemma 1. (Godsil and Royle [28]). Consider an n n symmetrical matrix A ¼ [aij] satisfying Pn a ¼ 0. The following statements hold: ij i¼1 (i) There exists an orthogonal matrix U, such that 0 0 U T AU ¼ , 0 n where * represents any nonzero matrix with an appropriate dimension and 0 represents any zero matrix with an appropriate dimension; (ii) Matrix A has a zero eigenvalue with the eigenvector 1 ¼ ð1,1,. . .,1Þ 2 Rn :
3. Main results In this section, the group consensus problem for multi-agent system (1) with the hybrid protocol (3) is investigated. The convergence analysis is presented and algebraic criterions are established in the couple-group average-consensus problem. Furthermore, the multigroup consensus as an extension of the couple-group consensus is also discussed. Given the protocol (3), the multi-agent system (1) evolves according to the following dynamic: 8 P P aij ðxj xi Þ þ bij xj ðhk Þ 8i 2 I1 , > > < vj 2N1i vj 2N2i P P ð4Þ x_ i ðtÞ ¼ , 8t 2 ½hk ,hkþ1 Þ, k 2 N þ , 8i 2 I2 , bij xj ðhk Þ þ aij ðxj xi Þ > > : vj 2N1i vj 2N2i we use vector notations to rewrite the equations in Eq. (4) with a compact form as follows: 2 3 2 3 2 3 x_ 1 x1 x1 ðhk Þ 6 x_ 7 6 x 7 6 x ðh Þ 7 6 2 7 6 2 7 6 2 k 7 6 6 7 7 6 7 6 ^ 7 7 " #6 ^ 7 " #6 ^ 6 7 6 7 6 7 0 B L 0 1 1 6 x_ 7 6 x 7 6 x ðh Þ 7 6 n 7¼ 6 n 7þ T 6 n k 7, 8t2 ½hk ,hkþ1 Þ, k 2 N þ , B1 0 6 0 L2 6 6 7 7 7 6 x_ nþ1 7 6 xnþ1 7 6 xnþ1 ðhk Þ 7 6 7 6 7 6 7 6 ^ 7 6 ^ 7 6 7 ^ 4 5 4 5 4 5 xnþm ðhk Þ x_ nþm xnþm ð5Þ where L1ARn n and L2ARm m are the Laplacian matrices of network G1 and G2, respectively, and the matrix B1ARn m describes the information exchange between different groups.
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Define
"
L1 L¼ 0
x ¼ ðx1 ,x2 ,. . .,xnþm ÞT ,
# 0 , L2
" B¼
0
B1
BT1
0
581
# ,
and system (5) can be rewritten as _ ¼ LxðtÞ þ Bxðhk Þ, xðtÞ
8t 2 ½hk ,hkþ1 Þ,
k 2 N þ:
ð6Þ
By the theory of ordinary differential equations [32], the solution of Eq. (6) in tA[hk,hkþ1) is Z t eLðttÞ Bxðhk Þ dt xðtÞ ¼ eLðthk Þ xðhk Þ þ hk Z t Lðthk Þ LðttÞ ¼ e þ e B dt xðhk Þ, 8t 2 ½hk ,hkþ1 Þ: ð7Þ hk
When t¼ hkþ1, by the continuity of the solution, we can obtain that Z hkþ1 xðhkþ1 Þ ¼ eLðhkþ1 hk Þ þ eLðhkþ1 tÞ B dt xðhk Þ:
ð8Þ
hk
Define x(hkþ1)¼ x(kþ1), x(hk)¼ x(k) Z hkþ1 CðkÞ ¼ eLðhkþ1 hk Þ þ eLðhkþ1 tÞ B dt hk
and rewrite Eq. (8) as xðk þ 1Þ ¼ CðkÞxðkÞ:
ð9Þ
The following lemma claims that the study of system (9) is sufficient for the investigation of system (6). Lemma 2. System (6) solves the couple-group average-consensus problem if and only if system (9) solves the couple-group average-consensus problem. Proof. The necessity follows from the definition of the state variable x(k). To prove the sufficiency, assume that system (9) solves the couple-group averageconsensus problem. P P Let x * ¼ aU1nþb1m, where a ¼ 1=n ni ¼ 1 xi ð0Þ, b ¼ 1=m nþm i ¼ nþ1 xi ð0Þ such that limk-1 x ðkÞ ¼ xn . Obviously, one has Z t eLðthk Þ þ eLðttÞ B dt xn ¼ xn , CðkÞxn ¼ xn , hk
for any ðk,tÞ 2 N þ ½hk ,hkþ1 Þ: In fact, by the definition of C(k), it follows: Z hkþ1 CðkÞxn ¼ ½eLðhkþ1 hk Þ þ eLðhkþ1 tÞ B dtxn hk
¼e
Lðhkþ1 hk Þ n
Z
hkþ1
x þ hk
eLðhkþ1 tÞ Bxn dt:
ð10Þ
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By the definition of B and (A3), one obtains that Bx * ¼ 0, hence, Z hkþ1 eLðhkþ1 tÞ Bxn dt ¼ 0: hk
Meanwhile, consider the term eLðhkþ1 hk Þ xn , it can be shown that " # 1 X 1 k k k Lðhkþ1 hk Þ n ð1Þ ðhkþ1 hk Þ L xn : x ¼ e k! k¼0
ð11Þ
Due to the fact that Lx * ¼ 0, one obtains that eLðhkþ1 hk Þ xn ¼ xn . Therefore, C(k)x * ¼ x * , and one also has Z t Lðthk Þ LðttÞ e þ e B dt xn ¼ xn : hk
For any given t40, there exists k, such that tA[hk,hkþ1), and Z t Lðth Þ n LðttÞ n k þ e B dt xðhk Þx jjxðtÞx jj ¼ e h Z kt Lðth Þ k ¼ e þ eLðttÞ B dt ðxðhk Þxn Þ h Z tk Lðth Þ k re þ eLðttÞ B dtjjUjjxðhk Þxn :
ð12Þ
hk
Because hkþ1 hk 2 ½t0 ,t1 , 8k 2 N þ , the norm of Z t eLðthk Þ þ eLðttÞ B dt hk
is bounded, namely, there exists a constant M40, such that Z t eLðttÞ B dtjjrM, 8k 2 N þ : jjeLðthk Þ þ hk
Since k-N as t-N, one has lim jjxðtÞxn jjrMU lim jjxðhk Þxn jj ¼ 0,
t-1
k-1
ð13Þ
which yields lim jjxðtÞxn jj ¼ 0:
t-1
ð14Þ
Thus, system (6) solves the couple-group average-consensus problem. This completes the proof. & By Lemma 2, system (9) can be regarded as a discrete-time multi-agent system consisting of nþm agents with time-varying parameters, which should be emphatically investigated.
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Let U1T ¼ ½u11 ,u12 2 Rnn and U2T ¼ ½u21 ,u22 2 Rmm be two orthogonal matrices, such that 2 3 2 3 0 0 6 7 6 7 lnþ2 l2 6 7 6 7 T T 6 7, ð15Þ U1 UL1 UU1 ¼ 6 7, U2 UL2 UU2 ¼ 6 7 & & 4 5 4 5 lnþm ln where 0ol2 rl3 r rln are the eigenvalues of L1, while 0olnþ2 rlnþ3 r rlnþm are the eigenvalues of L2, u11ARn 1, and u21ARm 1. According to Eq. (15), one has u11 ¼ pffiffiffiffi pffiffiffiffi pffiffiffi pffiffiffi ½1= n,. . .,1= nT , u21 ¼ ½1= m, ,1= mT , furthermore, u21ARn (n1), and each column vector of u21 represents an eigenvector of L1, while u22ARm (m1), and each column vector of u21 represents an eigenvector of L2. It is inferred from Eq. (15) that " # " #T " # " # " #T U1 0 U1 0 U1 0 L1 0 U1 0 ¼ ULU U U 0 U2 0 U2 0 U2 0 L2 0 U2 3 2 0 7 6 l2 7 6 7 6 7 6 & 7 6 7 6 ln 7 6 7: 6 ð16Þ ¼6 7 0 7 6 7 6 lnþ2 7 6 7 6 7 6 & 5 4 lnþm Denote " yðkÞ ¼
U1
0
0
U2
# xðkÞ,
then system (9) can be rewritten as yðk þ 1Þ ¼ C 0 ðkÞyðkÞ,
ð17Þ
where "
U1 C ðkÞ ¼ 0 0
# " 0 U1 UCðkÞU U2 0
0 U2
#T :
0
The following lemma claims that C (k) is a symmetric matrix on a joining condition. 0
Lemma 3. Given system (17), if L1UB1 ¼ B1UL2, then C (k) is a symmetric matrix. Proof. In fact, BT1 UL1 ¼ L2 UBT1 , hence, one has " # " # 0 B1 L2 0 L1 B1 BUL ¼ ¼ ¼ LUB, BT1 L1 0 L2 BT1 0
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584
namely, B and L are commutative matrices, which implies that B and eLt are commutative T matrices. According to (A4) and the fact that ðeLt ÞT ¼ eL t , it follows that T Z hkþ1 C T ðkÞ ¼ eLðhkþ1 hk Þ þ eLðhkþ1 tÞ B dt hk
T
Z
hkþ1
T
þ BT eL ðhkþ1 tÞ dt hk Z hkþ1 BeLðhkþ1 tÞ dt ¼ eLðhkþ1 hk Þ þ hk Z hkþ1 eLðhkþ1 tÞ B dt ¼ CðkÞ, ¼ eLðhkþ1 hk Þ þ ¼ eL
ðhkþ1 hk Þ
ð18Þ
hk 0
which shows that C (k) is a symmetric matrix. & Remark 3. The condition L1UB1 ¼ B1UL2 is called a joining condition between two different groups. On this condition, B and L are commutative matrices, which reflects equivalent relation between two different groups. Furthermore, the following results are based on the joining condition. 0
Considering C(k), C (k), one obtains that " # " #T U1 0 U1 0 Lðhkþ1 hk Þ U ¼ diagf1,. . .,eln ðhkþ1 hk Þ , 1,. . .,elnþm ðhkþ1 hk Þ g, Ue 0 U2 0 U2 ð19Þ "
U1
0
0
U2 Z
# Z U
hkþ1
¼ hkþ1
¼ hk
U1 0
hk
Z
e
hk
"
"
"
hkþ1
U1 0
Lðhkþ1 tÞ
B dtU
U1
0
#T
U2 " #T 0 U1 0 Lðhkþ1 tÞ dtUB Ue U2 0 U2 # " #T " U1 0 U1 0 Lðhkþ1 tÞ U dtU Ue U2 0 U2 0 0
#
# " 0 U1 UBU U2 0 lnþm ðhkþ1 hk Þ
1eln ðhkþ1 hk Þ 1e ¼ diag hkþ1 hk ,. . ., , hkþ1 hk ,. . ., ln lnþm " # T 0 U1 B 1 U2 : U2 BT1 U1T 0 Therefore, according to (A3), one has " T # " u11 0 T U1 B1 U2 ¼ T UB1 U½u21 ,u22 ¼ u12 0 where P1 ¼ uT12 B1 u22 2 Rðn1Þðm1Þ :
0 P1
0 U2
#T
ð20Þ
# ,
ð21Þ
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Denote
C1 ðkÞ ¼ diag el2 ðhkþ1 hk Þ ,. . .,eln ðhkþ1 hk Þ ,
l ðh h Þ C2 ðkÞ ¼ diag e nþ2 kþ1 k ,. . .,elnþm ðhkþ1 hk Þ , 1el2 ðhkþ1 hk Þ 1eln ðhkþ1 hk Þ ,. . ., , D1 ðkÞ ¼ diag l2 ln 1elnþ2 ðhkþ1 hk Þ 1elnþm ðhkþ1 hk Þ ,. . ., : D2 ðkÞ ¼ diag lnþ2 lnþm 0
Due to the symmetry of C (k), one has D2 ðkÞPT1 ¼ PT1 D1 ðkÞ, and then system (17) could be rewritten as 2 3 1 0 0 0 6 7 C1 ðkÞ 0 D1 ðkÞP1 7 60 6 7yðkÞ: ð22Þ yðk þ 1Þ ¼ 6 7 0 1 0 40 5 0 PT1 D1 ðkÞ 0 C2 ðkÞ The following theorem gives a necessary and sufficient condition for the couple-group average-consensus. Theorem 1. System (6) solves the couple-group average-consensus problem if and only if " C1 ðtÞ jjW ðtÞjj ¼ T P1 D1 ðtÞ
D1 ðtÞP1 C2 ðtÞ
# o1,
8t 2 ½t0 ,t1 ,
ð23Þ
where
C1 ðtÞ ¼ diag el2 t ,. . .,eln t , C2 ðtÞ ¼ diag elnþ2 t ,. . .,elnþm t , 1el2 t 1eln t ,. . ., : D1 ðtÞ ¼ diag l2 ln
Proof. To prove the sufficiency, we first prove that system (9) solves the couple-group average-consensus problem. Let 2 3 1 0 0 0 60 0 1 0 7 6 7 Zðk þ 1Þ ¼ 6 7yðk þ 1Þ, 0 5 4 0 In1 0 0 0 0 Im1 then
2
1
0
0
6 60 Zðk þ 1Þ ¼ 6 60 4 0
1 0
0 C1 ðkÞ
0
PT1 D1 ðkÞ
0
3
2 1 7 0 7 7zðkÞ ¼ 6 4 D1 ðkÞP1 7 5 C2 ðkÞ
3 1 W ðkÞ
7 5zðkÞ
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2
3
1
6 6 ¼6 6 4
1 k Y
7 7 7Zð0Þ, 7 W ðiÞ 5
ð24Þ
i¼0
Hence, 2
2
3
Z1 ðk þ 1Þ 6 6 6 Z ðk þ 1Þ 7 lim 4 2 5 ¼ lim 6 k-1 k-16 4 Z3 ðk þ 1Þ
3 Z1 ð0Þ 7 7 Z1 ð0Þ 6 Z ð0Þ 7 76 6 (2 7 ) 7: 7 6 Z ð0Þ ¼ lim 2 k 4 5 k
Y Y 7 7 6 k-14 W ðiÞ 5 Z ð0Þ W ðiÞ Z3 ð0 5 3
1 1
2
2
3
3
i¼0
i¼0
ð25Þ Due to the continuity of :W(t):, there exists t0A[t0,t1], such that jjW ðt0 Þjj ¼ maxt2½t0 ,t1 jjW ðtÞjj. Denote M0 ¼ :G(t0:, and thus it follows from the condition (23) that M0o1. Therefore, ( ) Y Y k k lim jjZ3 ðk þ 1Þjj ¼ lim W ðiÞ Z3 ð0Þr lim W ðiÞUjjZ3 ð0Þjj k-1 k-1 k-1 i¼0 i¼0 r lim M0kþ1 UjjZ3 ð0Þjj ¼ 0:
ð26Þ
k-1
Namely, limk-1 Z3 ðk þ 1Þ ¼ 0. According to (25), one has 3 2 3 2 3 2 Z1 ðk þ 1Þ y1 ð0Þ Z1 ð0Þ 7 6 7 6 7 6 lim 4 Z2 ðk þ 1Þ 5 ¼ 4 Z2 ð0Þ 5 ¼ 4 y3 ð0Þ 5, k-1 Z3 ðk þ 1Þ 0 0 2
ð27Þ
1
0
0
0
32
60 6 lim yðk þ 1Þ ¼ 6 40 k-1 0
0 1
In1 0
0 0
76 y3 ð0Þ 7 6 07 76 7 6 7 76 7¼6 7, 54 0 5 4 y3 ð0Þ 5
0
0
Im1
" lim xðk þ 1Þ ¼
k-1
U1
0
0
U2
#T
2
3 y1 ð0Þ 6 0 7 6 7 U6 7 4 y3 ð0Þ 5 0
y1 ð0Þ
0
3
2
y1 ð0Þ
0
3 ð28Þ
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! 3 n 1 X 3 6 2 xi ð0Þ 7 1 7 6 n i¼1 pffiffiffi y1 ð0Þ 7 6 7 6 6 n 7 7 6 6 ^ 7 7 6 6 ! 7 ^ 7 6 6 n 7 7 6 1 X 6 1 7 6 pffiffiffi y ð0Þ 7 6 x ð0Þ 7 i 7 6 1 n 7 6 7 6 6 n i¼1 7 6 !7 ¼ ¼6 1 7: 6 7 nþm X 7 1 6 pffiffiffiffi y3 ð0Þ 7 6 6 xi ð0Þ 7 7 6 6 m 7 7 6m 6 7 i ¼ nþ1 7 6 6 7 ^ 7 6 6 7 ^ 7 6 6 5 6 4 1 !7 7 nþm pffiffiffiffi y3 ð0Þ X 7 61 m 4 xi ð0Þ 5 m i ¼ nþ1
587
2
ð29Þ
Thus, it is proved that system (9) solves the couple-group average-consensus problem. Therefore, according to Lemma 2, system (6) solves the couple-group average-consensus problem. Next, we apply the reduction to absurdity to prove the necessity. Suppose that Eq. (23) is 0 not true, then there exists t A[t0,t1], such that :W ðt0 Þ:Z1. Therefore, there also exists an 0 eigenvector Z , such that :Z 0 : ¼ 1 and :W ðt0 Þz0 :Z1. From Eq. (26), if one chooses 0 0 Z3(0) ¼ Z and hkþ1hk ¼ t for all k, then Z3(kþ1) could not converge to zero as k-N. As a result, system (9) could not solve the couple-group average-consensus problem. According to Lemma 2, system (6) also could not solve the couple-group averageconsensus problem, and this results in a contradiction. This completes the proof. & In Theorem 1, the condition (23) is a necessary and sufficient condition. However, it is difficult to be implemented in practical applications, because we should verify that every point in [t0,t1] satisfies the condition (23). On the other hand, since the matrices L1, L2 are given in the practical situation, we should design appropriate parameters t0, t1, and matrix B such that system (6) solves the couple-group average-consensus problem. In view of these points, we give the algebraic criteria about the couple-group average-consensus problem in Theorem 2.
Theorem 2. Given system (6), if rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :B:F 1 n þ m2 , o 2 n þ m3 l2 then there exists T0 ¼ T0(n, m, L1, L2), such that for all t0A[T0,N), system (6) solves the couple-group average-consensus problem. Proof. Consider the auxiliary function rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 jtrW ðtÞj2 , gðtÞ ¼ jjC1 ðtÞjj2F þ jjC2 ðtÞjj2F þ 2jjD1 ðtÞP1 jj2F n þ m2
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Then n þ m3 2 jtrW ðtÞj2 1 jtrW ðtÞj2 2 g ðtÞ ð n þ m3 Þg ¼ ð t Þ n þ m2 n þ m2 n þ m2 ðn þ m2Þ2 1 ðn þ m3Þ jjC1 ðtÞjj2F þ jjC2 ðtÞjj2F þ 2jjD1 ðtÞP1 jj2F jtrW ðtÞj2 n þ m2 h 2 1 ðn þ m3Þ jjC1 ðtÞjj2F þ jjC2 ðtÞjj2F trW ðtÞ ¼ n þ m2 þ2ðn þ m3ÞjjD1 ðtÞP1 jj2F ,
¼
ð30Þ
Because of jtrW ðtÞj2 rðn þ m2Þ½jjC1 ðtÞjj2F þ jjC2 ðtÞjj2F , one has n þ m3 2 jtrW ðtÞj2 1 g ðtÞ 2ðn þ m3ÞjjD1 ðtÞP1 jj2F Z 2 n þ m2 n þ m2 ðn þ m2Þ jjC1 ðtÞjj2F jjC2 ðtÞjj2F :
ð31Þ
Moreover, 1 2ðn þ m3ÞjjD1 ðtÞP1 jj2F jjC1 ðtÞjj2F jjC2 ðtÞjj2F t-þ1 n þ m2 2ðn þ m3Þ jjD2 ð1ÞP1 jj2F 40: ¼ n þ m2 lim
ð32Þ
According to Eqs. (31) and (32), there exists T00 ¼ T00 ðn,m,L1 ,L2 Þ such that for all t 2 ½T00 ,1Þ n þ m3 2 jtrW ðtÞj2 g ðtÞ4 : n þ m2 ðn þ m2Þ2
ð33Þ
By the property of Frobenius norm, it is easy to obtain that " # 0 0 2 1 jjP1 jj2F ¼ ¼ jjU1 B1 U2T jj2F ¼ jjB1 jj2F ¼ jjBjj2F , 0 P1 2
ð34Þ
F
lim g2 ðtÞ ¼ lim jjC1 ðtÞjj2F þ jjC2 ðtÞjj2F þ 2jjD1 ðtÞP1 jj2F
t-þ1
t-þ1
¼ 2jjD2 ð1ÞP1 jj2F r
2jjP1 jj2F jjBjj2F ¼ : l22 l22
From Eq. (35) and the condition rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :B:F 1 n þ m2 , o 2 n þ m3 l2
1 jtrW ðtÞj2 n þ m2
ð35Þ
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589
there exists T 00 0 ¼ T 00 0 ðn,m,L1 ,L2 Þ such that for all t 2 ½T 00 0 ,1Þ the following inequality hold n þ m3 2 1 g ðtÞo : n þ m2 4
ð36Þ
Let T0 ¼ maxfT00 ,T 00 0 g, for all tA[T0,N), one has rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi trW ðtÞ n þ m3 gðtÞo1: þ n þ m2 n þ m2
ð37Þ
Now, it follows from Theorem 1 in [33] that for t0A[T0,N), the eigenvalues of G(t) belong to (1,1), namely " # C1 ðtÞ D1 ðtÞP1 :W ðtÞ: ¼ T o1, 8t 2 ½t0 ,t1 : C2 ðtÞ P1 D1 ðtÞ By applying Theorem 1, system (6) solves the couple-group average-consensus problem. This completes the proof. & Remark 4. In Theorem 2, the algebraic criterion rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :B:F 1 n þ m2 o 2 n þ m3 l2 is simple to implement, and the parameter T0 is related to the communication topology and the number of agents in different groups. If we consider the information exchanges from external groups as disturbances, it can be inferred from Theorem 2 that the couplegroup average-consensus of multi-agents system can still be solved when the disturbances are sufficiently small and the intervals of the update time are adequately long. Due to the sufficiency of the criteria, there may exist certain extent of conservatism in the selection of T0. Obviously, we have T0a0, in fact, if T0 ¼ 0, W(0) is an identity matrix, namely, :W ðT0 Þ: ¼ 1, which does not satisfy condition (23). In the next section, it can be found that T0 is close to 0, which indicates that less conservative results can be obtained. The following corollary of Theorem 1 gives the convergence rate of hybrid protocol in group consensus problem. Corollary 3. Given system (9), jjxðk þ 1Þxn jjrM1kþ1 Ujjxð0Þjj, where 1elt0 1elt1 lt0 lt1 jjBjj, e jjBjj , þ þ M1 ¼ max e l l with l ¼ minfl2 ,lnþ2 g. Furthermore, if 8k 2 N þ ,
hkþ1 hk ¼ T0 , then
kþ1 1elT0 jjBjj Ujjxð0Þjj: jjxðk þ 1Þxn jjr elT0 þ l
ð38Þ
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Proof. Since " yðkÞ ¼
U1
0
0
U2
"
U1 y ¼ 0 n
2
# xðkÞ,
1
0
60 0 6 Zðkþ1Þ ¼ 6 4 0 In1 0 0
0
0
3
1 0 7 7 7yðkþ1Þ, 0 0 5 0 Im1
# 0 xn U2
and 2
1
0
60 0 6 Zn ¼ 6 4 0 In1 0 0
0
0
3
1 0
0 0
7 7 n 7y , 5
0
Im1
one has jjxðk þ 1Þxn jj ¼ jjyðk þ 1Þyn jj ¼ jjZðk þ 1ÞZ n jj 2 3 1 2 3 1 6 7 1 6 7 6 7 6 7 1 ¼ 6 k 5Zð0Þ Y 7Zð0Þ4 4 W ðiÞ 5 0 i¼0 2 3 0 " # 6 7 k 0 Y 6 7 7UjjZð0Þjjr r6 rðW ðiÞÞ jjxð0Þjj, k Y 6 7 4 5 i¼0 W ðiÞ i¼0 Then considering r(W(t)) with tA[t0,t1], one obtains " C1 ðtÞ rðW ðtÞÞ ¼ T P1 D1 ðtÞ
D1 ðtÞP1 C2 ðtÞ
#
2 3 2 3 0 0 # " # 6 7 6 7 " 0 B 1 C1 ðtÞ D1 ðtÞ 6 7 6 7 U1 7þ6 7U ¼ 6 U T 7 7 6 6 B1 0 0 0 U2 4 5 4 5 C2 ðtÞ D2 ðtÞ 2 3 2 3 0 0 # 6 7 6 7 " D1 ðtÞ C1 ðtÞ 6 7 6 7 0 B1 7 7 6 6 r 6 7 þ 6 7U BT 0 0 0 1 4 5 4 5 C2 ðtÞ D2 ðtÞ
ð39Þ
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" # " # " C1 ðtÞ D1 ðtÞ 0 ¼ þ U C2 ðtÞ D2 ðtÞ BT1 Let l ¼ minfl2 ,lnþ2 g, and " # C1 ðtÞ lt re , C2 ðtÞ
# B1 : 0
591
ð40Þ
" # D1 ðtÞ rð1elt =lÞ, D2 ðtÞ
then rðW ðtÞÞrelt þ
1elt jjBjj: l
ð41Þ
0
Denote f(t)¼ eltþ((1elt)/l)99B99, it follows f (t)¼ [99B99l] elt, hence, f(t) is a monotonic function, let M1 ¼ maxt0 rtrt1 f ðtÞ ¼ maxff ðt0 Þ,f ðt1 Þg, then r(W(t))rM1. According to Eq. (39), one has jjxðk þ 1Þxn jjrM1kþ1 Ujjxð0Þjj,
ð42Þ
Moreover, if 8k 2 N þ , hkþ1hk ¼ T0, then M1 equals to elT0 þ ð1elT0 =lÞ:B:. This completes the proof. & Remark 5. According to Corollary 1, the convergence rate of group consensus depends on the parameters l, t0, t1 and B. Therefore, this property can be improved by selecting proper values of these parameters. Obviously, the result in [10] is a special case of the result (38). In fact, if :B: ¼ 0, the information exchange between different groups does not exist, Eq. (38) is then rewritten as jjxðk þ 1Þxn jjrelðkþ1ÞT0 Ujjxð0Þjj, which is consistent with the result in [10].
Furthermore, the couple-group average-consensus problem can be extended to a more general case, i.e. the agents in a network reach more than two consistent states asymptotically. Without loss of generality, suppose that a complex network (G,x) is composed of m subnetworks, and the qth sub-network has nq agents with a corresponding topology graph Gq, q ¼ 1,2,. . .,m, then a multi-group consensus can be defined as limt-1 jjxi xj jj ¼ 0, 8i, jAGq, q ¼ 1,2,. . .,m. For the complex network (G,x), assume that the topology graph G is undirected, and then a new hybrid protocol can be defined as follows: X X ui ðtÞ ¼ bij xj ðhk Þ þ þ aij ðxj xi Þ þ vj 2N1i
þ
X
vj 2Nqi
bij xj ðhk Þ,
8i 2 Gq ,
t 2 ½hk ,hkþ1 Þ,
k 2 Nþ
vj 2Nmi
where the sequence{hk} satisfies the assumptions (A1) and (A2).
ð43Þ
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Given the hybrid protocol (43), the multi-agent system (1) evolves according to the following system: X X bij xj ðhk Þ þ þ aij ðxj xi Þ þ x_ i ðtÞ ¼ vj 2 N1i
þ
vj 2 Nqi
X
bij xj ðhk Þ,
8i 2 Gq ,
t 2 ½hk ,hkþ1 Þ,
k 2 N þ,
ð44Þ
vj 2 Nmi
Then Eq. (44) can be translated into a matrix form _ ¼ LxðtÞ þ Bxðhk Þ, xðtÞ where
2
6 6 L¼6 6 4
8t 2 ½hk ,hkþ1 Þ,
k 2 N þ,
ð45Þ
3
L1 L2 & Lp
7 7 7, 7 5
with Lq 2 Rnq nq being the Laplacian matrix of sub-network Gq, and the matrix 3 2 0 B12 B1m 7 6 T 0 B2m 7 6 B12 7 B¼6 6 ^ ^ & ^ 7 5 4 BT1m
BT2m
0
describes the information exchanges among different groups. For instance, Bqi 2 Rnq ni represents the information exchanges between the qth group and the ith group. Here Bqi also satisfies the assumption (A3), i.e. 1T UBqi ¼ 0,
Bqi U1 ¼ 0:
ð46Þ
The following two results are generalizations of Lemmas 2 and 3. Lemma 4. System (45) solves the multi-group average-consensus problem if and only if system (9) solves the multi-group average-consensus problem. 0
Lemma 5. Given system (17), if LqUBqi ¼ BqiULi, 8q, i, then C (k) is a symmetric matrix, where 2 3 2 3T U1 U1 6 7 6 7 U2 U2 6 7 6 7 C 0 ðkÞ ¼ 6 7UCðkÞU6 7 , & & 4 5 4 5 Um Um 3 2 0 7 6 lðqÞ 7 6 2 7 Uq ULq UUqT ¼ 6 7 6 & 5 4 ðqÞ ln q ðqÞ ðqÞ and 0olðqÞ 2 rl3 r rlnq are eigenvalues of Lq.
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Remark 6. The condition Lq Bqi ¼ Bqi Li, 8q, i is called a joining condition between different groups, which means that different groups are equivalent. Under this condition, both B and L are commutative matrices. Furthermore, based on the joining condition, the following results are obtained. pffiffiffiffiffi pffiffiffiffiffi Obviously, the first column vectors of UqT is uq1 ¼ ½ð1= nq Þ,. . .,ð1= nq ÞT . According to Eq. (44), we have " T # " # uq1 0 0 T Uq Bqi Ui ¼ T UBqi U½ui1 ,ui2 ¼ , 0 Pqi uq2 where Pqi ¼ uTq2 Bqi ui2 2 Rðnq 1Þðni 1Þ . The following result is a generation of Theorem 1. Theorem 3. System (45) solves the multi-group average-consensus problem if and only if 2 C1 ðtÞ D1 ðtÞP12 6 T C2 ðtÞ 6 P12 D1 ðtÞ jjW ðtÞjj ¼ 6 6 ^ ^ 4 T P1m D1 ðtÞ PT2m D2 ðtÞ
3 D1 ðtÞP1m 7 D2 ðtÞP2m 7 7o1, 7 & ^ 5 Cm ðtÞ
8t 2 ½t0 ,t1 ,
ð47Þ
where n ðqÞ o ðqÞ Cq ðtÞ ¼ diag el2 t ,. . .,elnq t ,
( Dq ðtÞ ¼ diag
ðqÞ
1el2 lðqÞ 2
ðqÞ
t
,. . .,
1elnq t lðqÞ nq
) ,
8q 2 f1,2,. . .,mg: In Theorem 3, the condition (47) is a necessary and sufficient condition. However, it is difficult to implement because we should verify that every point in [t0,t1] satisfies the condition (47). Therefore, inspired by the above analysis, the following theorem gives the algebraic criteria about the multi-group average-consensus problem, and the proof of which is similar to Theorem 2. Theorem 4. Given system (45), if !ð1=2Þ Pm jjBjjF 1 q ¼ 1 nq m Pm o , 2 l q ¼ 1 nq m1 ðmÞ where l ¼ minflð1Þ 2 ,. . .,l2 g, there exists T0 ¼ T0 ðn1 ,. . .,nm ,L1 ,. . .,Lm Þ, such that for all t0A[T0,N), system (45) solves the multi-group average-consensus problem.
4. Simulation results In this section, an example is provided to illustrate the effectiveness of the proposed theoretical results. We consider five agents with the interaction topology given in Fig. 2.
594
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1
1/3 G1
7/8 -7/8
1/3 -7/16 7/16
2
4
1/2
-7/16
G2
5 3
7/16
Fig. 2. The interaction topology.
From Fig. 2, the Laplacian matrixes of G1 and G2 can be easily obtained as follows: 3 2 2 1 1 2 3 6 3 1 1 3 37 7 6 7 7 6 1 1 6 27 7, L2 ¼ 6 2 0 L1 ¼ 6 7 6 3 3 4 1 1 5 7 6 4 1 1 5 2 2 0 3 3 and the eigenvalues of L1 and L2 are l1 ¼ 0, l3 ¼ 1, l4 ¼ 0, l5 ¼ 1. The information exchange between G1 and G2 is 3 2 7 7 6 8 87 7 6 6 7 7 7 7 6 B1 ¼ 6 7 6 16 16 7 4 7 7 5 16 16 According to Eqs. (31) and (34) in Theorem 2, it can be concluded that system (6) solves the couple-group average-consensus problem for all t0A[0.2959,N). We choose t0 ¼ 0.3, t1 ¼ 0.4, and the stochastic sampling intervals are shown in Fig. 3. With the initial T state xð0Þ ¼ 0:1 0:2 0:3 0:4 0:5 ,the couple-group average-consensus is reached asymptotically, and the state trajectories of five individual agents are shown in Fig. 4. 5. Conclusion A group consensus problem with discontinuous information transmission between different groups of dynamic agents was investigated. In the presence of undirected information exchange, this paper provided sufficient and necessary conditions for solving the couple-group average-consensus problem asymptotically by using the hybrid protocol. Moreover, by employing the tools of algebraic graph theory, matrix theory, and control theory, we performed the convergence analysis and established several algebraic criterions for the group consensus problem. In addition, as an extension of the couple-group consensus problem, the multi-group consensus problem was introduced and the corresponding analysis
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595
0.42
0.4
Sampling Intervals
0.38
0.36
0.34
0.32
0.3
0
5
10
15
20
25
30
Time
Fig. 3. The stochastic sampling intervals.
agent 1 agent 2 agent 3 agent 4 agent 5
0.6
States of agents
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
30
Time
Fig. 4. State trajectories of five individual agents.
results were derived by the similar techniques. Finally, the simulation confirmed the theoretical results. Further extensions will focus on disturbance rejection and robustness properties of the hybrid protocol, as well as switching network topologies between agents, communication dropouts and message transmission delay, and so on. Acknowledgment This work was supported by the National Natural Science Foundation of China under the Grants nos. 60974017, 61273212, and 61004097.
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