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Couple-group consensus for multi-agent networks of agents with discrete-time second-order dynamics
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Journal of the Franklin Institute 350 (2013) 3277–3292 www.elsevier.com/locate/jfranklin
Couple-group consensus for multi-agent networks of agents with discrete-time second-order dynamics Yuanzhen Fenga,b, Junwei Luc, Shengyuan Xua,n, Yun Zoua a
School of Automation, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, PR China b Department of Basic Science, Nanjing College for Population Program Management, Nanjing 210042, Jiangsu, PR China c School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042, Jiangsu, PR China Received 7 December 2012; received in revised form 1 July 2013; accepted 4 July 2013 Available online 8 August 2013
Abstract This paper considers the couple-group consensus problem for multi-agent networks with fixed and directed communication topology, where all agents are described by discrete-time second-order dynamics. Consensus protocol is designed such that some agents in a network reach a consistent value, while other agents reach another consistent value. The convergence of the system matrix is discussed based on the tools from matrix theory. An algebraic condition is established to guarantee couple-group consensus. Moreover, for a given communication topology, a theorem is derived on how to select proper control parameters and sampling period for couple-group consensus to be reached. Finally, simulation examples are presented to validate the effectiveness of the theoretical results. & 2013 Published by Elsevier Ltd. on behalf of The Franklin Institute
1. Introduction Recent years have witnessed the growing interest in cooperative control of multi-agent systems due to its wide applications in synchronization of coupled oscillators, rendezvous in space, and other areas. In these applications, the states of all agents are required to reach an agreement on some consistent value, which is known as the consensus problem. Up to now, many theoretical results have been achieved on consensus problems for multiagent systems [1–16]. A simple but compelling discrete-time model was proposed in [1], where a n
Corresponding author. Tel./fax: +86 25 84315463. E-mail addresses: [email protected], [email protected] (S. Xu).
0016-0032/$32.00 & 2013 Published by Elsevier Ltd. on behalf of The Franklin Institute http://dx.doi.org/10.1016/j.jfranklin.2013.07.004
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local control law was designed for all agents to move towards the same direction. Simulations were given to show that the headings of all agents will reach an agreement even if the neighbor sets of all agents may change as system evolves. By using the Lyapunov theory, consensus criteria were established in [3], under which multi-agent systems with switching topologies and time delays will reach average consensus asymptotically. In [7], the authors investigated the consensus problem for multi-agent systems with external disturbances and model uncertainty, where a model transformation was introduced and consensus conditions were given for multiagent systems to reach consensus with desired H 1 performance. Consensus protocols were designed and some necessary and/or sufficient conditions were established to guarantee the second-order consensus in [9]. By introducing generalized algebraic connectivity, sufficient conditions were derived in [13] for multi-agent systems with nonlinear dynamics to reach consensus. The second-order consensus problem for discrete-time multi-agent systems was addressed in [15,16], respectively. Note that the above references all consider such consensus where the states of all agents converge to the same consensus value. Due to the changes of situations or cooperative tasks, the consensus values may be different for agents from different sub-networks [17–19]. Group consensus, which includes the aforementioned consensus as a special case, was first introduced in [17] to represent such consensus where the states of all agents in the same sub-network reach the same consistent value while there is no agreement among different sub-networks. Average group consensus problems for multi-agent systems with undirected and balanced topologies were investigated in [17,18], respectively, where all agents have first-order dynamics. By applying double-tree-form transformation and Lyapunov direct method, the work in [17,18] was extended to the case of networks with switching topologies and time delays in [19]. Couple-group consensus of high-order multi-agent systems was addressed in [20], where couple-group consensus was guaranteed by the Hurwitz stability of the coefficient matrix. In this paper, we follow the work of [17,18,21] to study the couple-group consensus problem for multi-agent systems with fixed communication topology. Different from [17–19], all agents considered here have discrete-time second-order dynamics. As a comparison, a necessary and sufficient condition was given in [20] to ensure couple-group consensus of high-order multiagent systems. However, how to select proper control gains still remains unknown for the consensus conditions in [20] to be satisfied. The group consensus problem for multi-agent systems with continuous-time second-order dynamics was considered in [21]. The main contribution of this paper is as follows. Distributed consensus protocol is designed and some necessary and sufficient conditions are established for multi-agent systems to reach couple-group consensus. Furthermore, for a given communication topology, we provide an efficient way on how to select appropriate control parameters and sampling period for couple-group consensus to be achieved, which shows that couple-group consensus will be reached only if the real parts of the non-zero eigenvalues of the Laplacian matrix are positive. The rest of this paper is organized as follows. In Section 2, some preliminaries and the problem formulation are introduced. Couple-group consensus protocol is proposed and the eigenvalues as well as the corresponding eigenvectors of the system matrix are analyzed in Section 3. Main results are given in Section 4. Simulation results are presented in Section 5 to demonstrate the effectiveness of the theoretical results. Section 6 concludes this paper with a discussion. Notation. The following notations will be used throughout the paper. Let In (On) be the n n unit (zero) matrix; Omn be the m n zero matrix and 1n (0n ) be the n-dimensional column vector with all elements being 1 (0). The subscript will be omitted if it is clear from the
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context. ReðλÞ and ImðλÞ denote the real and imaginary part of a complex number λ, respectively. Cnn represents the set of all n n complex matrices. triagfA1 ; A2 ; …; Am g is used to denote the block lower triangular matrix with diagonal blocks A1 to Am. The superscript T means the transpose of a column vector or a matrix. J J refers to the standard Euclidean norm of vectors. j j denotes the absolute value of a real number or the modulus of a complex number. 2. Preliminaries and problem formulation 2.1. Preliminaries In this subsection, we will review some basic concepts in graph theory and introduce some lemmas which will be used in this paper. Let G ¼ ðV; E; AÞ be a directed graph with a finite nonempty set of nodes V ¼ fv1 ; v2 ; …; vn g, a set of edges EDV V and a weighted adjacency matrix A. An edge eij ¼ ðvj ; vi Þ A E means that node vi can receive information from node vj. A ¼ ½aij nn is defined as aij a 0 if eij A E and aij ¼ 0 if eij 2 = E. Moreover, aii ¼ 0 is assumed for all i. A graph is called undirected if aij ¼ aji . The neighbor set of node vi is denoted by N i ¼ fvj jeij A Eg. For a given network system of dynamic agents, a directed graph G will be used to model the information communication among all agents. Lemma 1 (Horn and Johnson [22]). If λ1 ; …; λn are the eigenvalues of AðA A Cnn Þ, the determinant of A is λ1 ⋯λn . Especially, if A is a real matrix, there exists an invertible matrix P such that P1 AP ¼ triagfλ1 ; …; λn g: Lemma 2 (Parks and Hahn [23]). Given a complex coefficient polynomial of order two as follows: hðsÞ ¼ s2 þ ða1 þ ib1 Þs þ a0 þ ib0 ; pffiffiffiffiffiffiffi where i ¼ 1; a1 ; b1 ; a0 and b0 are real constants. Then, h(s) is stable if and only if a1 40 and a1 b1 b0 þ a21 a0 b20 40. 2.2. Problem formulation Suppose that the network system under consideration consists of n+m agents. Each agent is regarded as a node in a directed graph G. To analyze the couple-group consensus problem, without loss of generality, we divide the communication network into two sub-networks, where the first n agents belong to the first sub-network and the remaining m agents belong to the second sub-network. G1 and G2 are used to model the information communication of these two subnetworks. Suppose that each agent is described by the following dynamics: ξi ðk þ 1Þ ¼ ξi ðkÞ þ τζ i ðkÞ; ζ i ðk þ 1Þ ¼ ζ i ðkÞ þ τui ðkÞ;
i ¼ 1; 2; …; n þ m;
ð1Þ
where ξi ðkÞ A R, ζ i ðkÞ A R and ui ðkÞ AR are the position, velocity and control input of agent i at time instant kτ, respectively; τ is the sampling period. Denote ξ1 ðkÞ ¼ ½ξ1 ðkÞ; …; ξn ðkÞT ;
ξ2 ðkÞ ¼ ½ξnþ1 ðkÞ; …; ξnþm ðkÞT ;
ξðkÞ ¼ ½ξ1 ðkÞT ; ξ2 ðkÞT T ;
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ζ 1 ðkÞ ¼ ½ζ 1 ðkÞ; …; ζ n ðkÞT ; ζ 2 ðkÞ ¼ ½ζ nþ1 ðkÞ; …; ζ nþm ðkÞT ; ζðkÞ ¼ ½ζ 1 ðkÞT ; ζ 2 ðkÞT T ; ℓ1 ¼ f1; 2; …; ng; ℓ2 ¼ fn þ 1; n þ 2; …; n þ mg; ℓ ¼ ℓ1 [ ℓ2 ; V 1 ¼ fv1 ; …; vn g; V 2 ¼ fvnþ1 ; …; vnþm g; N i;1 ¼ fvj A V 1 jeij A Eg; N i;2 ¼ fvj A V 2 jeij A Eg: Then, V ¼ V 1 [ V 2 , N i ¼ N i;1 [ N i;2 and Gk ¼ ðV k ; E k ; Ak Þ, where E k ¼ feij ji; j A ℓk g and Ak inherit A, k ¼ 1, 2. Therefore, N i;k can been seen as the neighbor set of agent i in Gk , k¼ 1, 2. Note that E 1 [ E 2 is a subset of E as information transition exists not only among agents in the same sub-network but also among agents from different sub-networks, where E represents the set of edges corresponding to the information communication among all agents in the communication network. Definition 1. Multi-agent system (1) is said to achieve couple-group consensus asymptotically if for any initial conditions, we have lim jξi ðkÞξj ðkÞj ¼ 0
k-1
and
lim jζ i ðkÞζ j ðkÞj ¼ 0
k-1
8 i; jA ℓl ; l ¼ 1; 2:
Assumption 1. We make the following assumptions as in [17,18]: ðA1Þ
j ¼ nþm
∑ aij ¼ 0
j ¼ nþ1
8 iA ℓ1
ðA2Þ
j¼n
∑ aij ¼ 0
j¼1
8 iA ℓ2 :
These assumptions mean that the interaction between the two subgroups is balanced. The problem to be addressed in this paper is to design consensus protocol and establish conditions under which couple-group consensus can be achieved by applying proposed protocol. 3. Consensus protocol and spectrum analysis for the system matrix In this section, consensus protocol will be designed and the spectrum of the system matrix will be analyzed. 3.1. Consensus protocol To solve the couple-group consensus problem for multi-agent system (1), the following consensus protocol: 8 ∑ a ½αðξj ðkÞξi ðkÞÞ þ βðζ j ðkÞζ i ðkÞÞ þ ∑ aij ½αξj ðkÞ þ βζ j ðkÞ 8 iA ℓ1 > > < 8 vj A N i;1 ij 8 vj A N i;2 ui ðkÞ ¼ ∑ aij ½αξj ðkÞ þ βζ j ðkÞ þ ∑ aij ½αðξj ðkÞξi ðkÞÞ þ βðζ j ðkÞζ i ðkÞÞ 8 iA ℓ2 > > : 8 vj A N i;1 8 vj A N i;2 ð2Þ is proposed, where α40 and β40 are the control parameters to be determined; aij Z 0 for all i, jA ℓk ðk ¼ 1; 2Þ and aij A R for all i A ℓ1 , jA ℓ2 or iA ℓ2 , jA ℓ1 . By applying protocol (2), the dynamics of system (1) can be re-written as " # " # ξðk þ 1Þ ξðkÞ ¼Φ ; ð3Þ ζðk þ 1Þ ζðkÞ
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where Φ ¼ I þ τΨ with
" Ψ¼
ð4Þ
Onþm
I nþm
αL
βL
#
and the Laplacian matrix L ¼ ½lij is defined as 8 if ia j a > < ij nþm lij ¼ ∑ aij otherwise i; j ¼ 1; 2; …; n þ m: > : j ¼ 1; jai
ð5Þ
From (4), one gets λi ðΦÞ ¼ 1 þ τλi ðΨ Þ;
i ¼ 1; 2; …; 2ðn þ mÞ;
ð6Þ
where λi ðΦÞ and λi ðΨ Þ are the i-th eigenvalues of Φ and Ψ , respectively; and Φ has the same eigenvectors as Ψ . For the linear system (3), the eigenvalues of matrix Φ play key roles in the convergence analysis. In the following, we put emphasis on the analysis for the eigenvalues and corresponding eigenvectors or generalized eigenvectors of Ψ . Then, by Relations (4) and (6), it is easy to derive the eigenvalues and the eigenvectors of Φ. To proceed with our analysis, we first give a required lemma. Lemma 3 (Yu and Wang [17]). Under Assumption 1, 0 is an eigenvalue of L of geometric multiplicity at least two. Proof. The definition of L in (5) ensures the property that ∑nþm j ¼ 1 lij ¼ 0 for all i ¼ 1; …; n þ m. So, 0 is an eigenvalue of L. Furthermore, one can verify that Lqi ¼ 0 qi ;
i ¼ 1; 2;
where q1 ¼ ½1Tn ; 0Tm T ;
q2 ¼ ½0Tn ; 1Tm T :
ð7Þ
Thus, q1 and q2 are two linearly independent right eigenvectors of L corresponding to zero eigenvalues. This completes the proof. □ 3.2. Eigenvalues and eigenvectors of the system matrix To find out the eigenvalues of Ψ , one needs to solve its characteristic equation " #! λI nþm I nþm detðλI 2ðnþmÞ Ψ Þ ¼ det ¼ 0; αL λI nþm þ βL where detðÞ represents the determinant of a matrix, detðλIΨ Þ is the characteristic polynomial of Ψ . By doing some computations as in [9], we have detðλIΨ Þ ¼ detðλIðλI þ βLÞαLðIÞÞ ¼ detðλ2 I þ ðα þ λβÞLÞ for the fact that I and L are commutable.
ð8Þ
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Suppose that μi are the eigenvalues of L, then λ2 þ ðα þ λβÞμi are the eigenvalues of matrix λ I þ ðα þ λβÞL, i ¼ 1; …; n þ m. Therefore, we know from Lemma 1 that 2
nþm
detðλIΨ Þ ¼ detðλ2 I þ ðα þ λβÞLÞ ¼ ∏ ðλ2 þ ðα þ λβÞμi Þ;
ð9Þ
i¼1
which implies that λ2i;7 ðΨ Þ þ ðα þ λi;7 ðΨ ÞβÞμi ¼ 0; where λi;7 ðΨ Þ are the eigenvalues of Ψ corresponding to μi . Hence, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi βμi 7 β2 μ2i 4αμi λi;7 ðΨ Þ ¼ 2 and μi ¼
λ2i;7 ðΨ Þ α þ βλi;7 ðΨ Þ
ð10Þ
ð11Þ
ð12Þ
can be inferred from Eq. (10). From Eqs. (11) and (12), we know that μi ¼ 0 is equivalent to λi;7 ðΨ Þ ¼ 0. So, we have the following result about the relationship of the zero eigenvalues of L and Ψ . Lemma 4. Ψ has an eigenvalue 0 of multiplicity1 four if and only if L has an eigenvalue 0 of multiplicity two. Remark 1. From Eq. (6), Φ has an eigenvalue 1 of multiplicity four if and only if L has an eigenvalue 0 of multiplicity two. Similar to the analysis in [21], we set out to solve the eigenvectors and generalized eigenvectors of Ψ corresponding to zero eigenvalues. Let w ¼ ½wT1 ; wT2 T be a right eigenvector of Ψ corresponding to zero eigenvalues, where w1 ; w2 A Rnþm . Then, we have Ψ w ¼ 02ðnþmÞ , i.e. # " #" # " Onþm I nþm 0nþm w1 ¼ ; 0nþm αL βL w2 which implies w2 ¼ 0nþm and Lw1 ¼ 0nþm . From Lemma 3, there exist two linearly independent right eigenvectors w1;r and w3;r of Ψ corresponding to zero eigenvalues, where w1;r ¼ ½qT1 ; 0Tnþm T and w3;r ¼ ½qT2 ; 0Tnþm T with q1 ¼ ½1Tn ; 0Tm T and q2 ¼ ½0Tn ; 1Tm T . It can be verified that ðΨ 0 IÞw2;r ¼ w1;r and ðΨ 0 IÞw4;r ¼ w3;r , where w2;r ¼ ½0Tnþm ; qT1 T and w4;r ¼ ½0Tnþm ; qT2 T . Hence, w2;r and w4;r are two generalized right eigenvectors of Ψ corresponding to zero eigenvalues. Suppose that p1 and p2 are left eigenvectors of L corresponding to zero eigenvalues which satisfy pT1 q1 ¼ 1 and pT2 q2 ¼ 1. Similarly, it can be proved that w2;l ¼ ½0Tnþm ; pT1 T and w4;l ¼ ½0Tnþm ; pT2 T are two left eigenvectors of Ψ corresponding to zero eigenvalues. 1
In the sequel, for the sake of simplicity, we simply write algebraic multiplicity as multiplicity.
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w1;l ¼ ½pT1 ; 0Tnþm T and w3;l ¼ ½pT2 ; 0Tnþm T are two generalized left eigenvectors of Ψ corresponding to zero eigenvalues.
4. Main results In this section, necessary and sufficient conditions will be derived to solve the couple-group consensus problem for multi-agent system (3) with fixed communication topology. Theorem 1. By applying consensus protocol (2), multi-agent system (1) achieves couple-group consensus asymptotically if and only if Φ has exactly an eigenvalue 1 of multiplicity four and all the other eigenvalues lie inside the unit circle. Furthermore, we have the following results about the final consensus values: jξi ðkÞpT11 ξ1 ð0ÞpT12 ξ2 ð0ÞkpT11 ζ 1 ð0ÞkpT12 ζ 2 ð0Þj-0
8 iA ℓ1
jξi ðkÞpT21 ξ1 ð0ÞpT22 ξ2 ð0ÞkpT21 ζ 1 ð0ÞkpT22 ζ 2 ð0Þj-0 ζ i ðkÞ-pT11 ζ 1 ð0Þ þ pT12 ζ 2 ð0Þ 8i A ℓ1 ζ i ðkÞ-pT21 ζ 1 ð0Þ þ pT22 ζ 2 ð0Þ 8i A ℓ2
8 iA ℓ2
as k-1, where p1 ¼ ½pT11 ; pT12 T and p2 ¼ ½pT21 ; pT22 T are left eigenvectors of L associated with zero eigenvalues with pT1 q1 ¼ 1 and pT2 q2 ¼ 1 (p11 ; p21 A Rn ; p12 ; p22 A Rm ); q1 and q2 are defined in Eq. (7). Proof (Sufficiency) Let J be the Jordan canonical form of Φ, then there exists an invertible matrix P ¼ ½w1;r ; w2;r ; w3;r ; w4;r ; … such that P1 ΦP ¼ J; where " J¼
J1
O4ð2ðnþmÞ4Þ
Oð2ðnþmÞ4Þ4
J2
2
# ;
1 1 60 1 6 J1 ¼ 6 40 0 0 0
0 0 1 0
3 0 07 7 7; 15 1
J2 represents the Jordan block corresponding to the other eigenvalues of Φ. To facilitate our analysis, we partition P into block form such as P ¼ ½P1 ; P2 ; where P1 ¼ ½w1;r ; w2;r ; w3;r ; w4;r , P2 is the block composed of right eigenvectors or generalized right eigenvectors corresponding to the other eigenvalues of Φ. In a similar way, P1 will be partitioned into " # Q1 P1 ¼ ; Q2
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where
2
wT1;l
3
6 wT 7 6 2;l 7 7 Q1 ¼ 6 6 wT 7; 4 3;l 5 wT4;l Q2 is the block composed of left eigenvectors or generalized left eigenvectors corresponding to the other eigenvalues of Φ. Then, we have Φk ¼ P1 J k1 Q1 þ P2 J k2 Q2 ; where 2
1 60 6 P1 J k1 Q1 ¼ P1 6 40 0
k 0 1 0 0 1 0 0
ð13Þ 2 3 1n pT11 0 6 6 1m pT21 07 7 Q ¼ 7 1 6 6 O k5 4 O 1
1n pT12 1m pT22
k1n pT11 k1m pT21
O
1n pT11
O
1m pT21
3 k1n pT12 7 k1m pT22 7 7 1n pT12 7 5 1m pT22
and J2 satisfies lim J k2 ¼ O2ðnþmÞ4 :
k-1
When the communication topology is fixed, system (3) will evolve according to the following dynamics: " # " # ξðkÞ ξð0Þ k ¼Φ : ζðkÞ ζð0Þ Based on Eq. (13), we obtain that 2 32 1 3 3 2 ξ1 ðkÞ 1n pT11 1n pT12 k1n pT11 k1n pT12 ξ ð0Þ 6 2 7 6 7 T T T T 76 2 6 ξ ðkÞ 7 6 1m p21 1m p22 k1m p21 k1m p22 76 ξ ð0Þ 7 76 7 6 7 lim 6 6 1 76 6 1 7 O 1n pT11 1n pT12 7 k-1 4 ζ ðkÞ 5 4 O 54 ζ ð0Þ 5 2 ζ ðkÞ O O 1m pT21 1m pT22 ζ 2 ð0Þ 2 32 1 3 3 2 1n pT11 1n pT12 k1n pT11 k1n pT12 ξ ð0Þ ξ1 ð0Þ 6 2 7 6 7 7 6 k 6 ξ ð0Þ 7 6 1m pT21 1m pT22 k1m pT21 k1m pT22 76 ξ2 ð0Þ 7 76 7 6 7 ¼ lim Φ6 6 1 76 6 1 7 O 1n pT11 1n pT12 7 k-1 4 ζ ð0Þ 5 4 O 54 ζ ð0Þ 5 O O 1m pT21 1m pT22 ζ 2 ð0Þ ζ 2 ð0Þ 2 1 3 2 3 1n pT11 ξ1 ð0Þ þ 1n pT12 ξ2 ð0Þ þ k1n pT11 ζ1 ð0Þ þ k1n pT12 ζ2 ð0Þ ξ ð0Þ 6 2 7 6 7 T 1 T 2 T 1 T 2 1 p ξ ð0Þ þ 1 p ξ ð0Þ þ k1 p ζ ð0Þ þ k1 p ζ ð0Þ 6 6 7 7 ξ ð0Þ m m m m 21 22 21 22 7 6 7 ¼ 0: ¼ lim P1 J k1 Q1 6 1 T 1 T 2 6 6 7 7 k-1 1n p11 ζ ð0Þ þ 1n p12 ζ ð0Þ 4 ζ ð0Þ 5 4 5 1 2 2 T T 1m p21 ζ ð0Þ þ 1m p22 ζ ð0Þ ζ ð0Þ Therefore, couple-group consensus is achieved asymptotically.
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(Necessity) We know from Lemma 3 and Remark 1 that Φ has an eigenvalue 1 of multiplicity at least four. If the sufficient condition does not hold, Φ has at least five eigenvalues whose modulus are greater than or equal to 1. Thus, rank ðJ k Þ44 holds as k-1. From the proof of the sufficiency, it is obvious that couple-group consensus is achieved if and only if 2 3 1n pT 6 7 6 1m qT 7 7 lim Φk -6 6 1n wT 7; k-1 4 5 1m z T where p; q; w and z are arbitrary column vectors with appropriate dimensions. This implies that rank ðΦk Þr 4 as k-1. Note the fact that rank ðΦk Þ ¼ rank ðJ k Þ, a contrary is resulted in. □ Remark 2. The final consensus values are specified in this paper when couple-group consensus is achieved, which were not discussed in [20]. Remark 3. If the network under consideration is divided into multi sub-networks, consensus protocol (2) and algebraic condition in Theorem 1 can be extended for multi-group consensus to be achieved. We notice that the algebraic condition in Theorem 1 is not straightforward to be checked. Now, for a given communication topology, the following theorem is proposed for choosing appropriate control parameters and sampling period to ensure couple-group consensus. Theorem 2. By applying consensus protocol (2), multi-agent system (1) reaches couple-group consensus asymptotically if and only if L has exactly an eigenvalue 0 of multiplicity two and 8 β > > > 4τ; < α ð14Þ 4 Reðμi Þ 4 Im2 ðμi Þ > 2 2 2β > f τ; μ τ þ τ 40; ¼ ðβατÞ > i : α αjμi j2 jμi j4 where μi are the non-zero eigenvalues of L, i ¼ 3; …; n þ m. Before giving the proof for Theorem 2, we present the following preliminaries. From Lemma 1, L is similar to triagfμ1 ; μ2 ; …; μnþm g, where μ1 ¼ μ2 ¼ 0. Then, Ψ is similar to " # I nþm Onþm : αtriagfμ1 ; μ2 ; …; μnþm g βtriagfμ1 ; μ2 ; …; μnþm g By using some elementary similarity transformations, we know that Ψ is similar to (" # " #) 0 1 0 1 ; …; : triag αμnþm βμnþm αμ1 βμ1 Therefore, Φ is similar to (" # " 1 1 τ ; …; triag ατμ ατμ1 1βτμ1 nþm
τ 1βτμnþm
#) :
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Consider " B¼
1
τ
#
ατλ 1βτλ
ðjλj40Þ;
its characteristic polynomial is hðs; λÞ ¼ detðsIBÞ ¼ ðs1Þ2 þ βτλðs1Þ þ ατ2 λ: By applying the bilinear transformation s ¼ ðs þ 1Þ=ðs1Þ, we get sþ1 2 ; λ ¼ αðτjλjÞ2 ðs1Þ2 þ 2βτλj2 ðs1Þ þ 4λ: λðs1Þ h s1 Define
sþ1 ;λ λðs1Þ h β 2β 4 ReðλÞ 4 ImðλÞ s1 2 1 s þ 1 þ gðs; λÞ ¼ ¼ s þ 2 i ; 2 ατ ατ αðτjλjÞ2 αðτjλjÞ2 αðτjλjÞ 2
ð15Þ then, by Lemma 2, gðs; λÞ is stable if and only if 8 β > > 1 40; 2 > < ατ 2 β 2β 4 ReðλÞ 4 ImðλÞ 2 > > >4 1 1 þ 40: : ατ ατ αðτjλjÞ2 αðτjλjÞ2
ð16Þ
Now, we give the proof for Theorem 2. Proof. From Theorem 1, couple-group consensus will be achieved if and only if Φ has exactly an eigenvalue 1 of multiplicity four and 1 is the unique eigenvalue of maximum modulus. By Lemma 4 and above analysis, couple-group consensus will be achieved if and only if L has exactly an eigenvalue 0 of multiplicity two and hðs; μi Þ are Schur stable, i ¼ 3; …; n þ m. Note that hðs; λÞ is Schur stable if and only if gðs; λÞ is stable. From inequality (16), gðs; μi Þ are stable if and only if 8 β > > > 1 40 2 > < ατ 2 β 2β 4 Reðμi Þ 4 Imðμi Þ 2 > > > 4 1 þ 1 40 > : ατ ατ αðτjμi jÞ2 αðτjμi jÞ2 hold for i ¼ 3; …; n þ m. This completes the proof.
□
Remark 4. Compared with [15], the sampling period is considered in this paper, which increases the difficulty in choosing control parameters for couple-group consensus to be reached. For a given communication topology, one may first fix the values of α and β, then try to find some proper sampling period for inequality (14) to be satisfied.
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Note that inequality (14) is satisfied only if β 4τ α and τ2
2β 4 Reðμi Þ τþ 40; α αjμi j2
which imply that all the non-zero eigenvalues of the Laplacian matrix have positive real parts. Corollary 1. By applying consensus protocol (2), multi-agent system (1) reaches couple-group consensus asymptotically only if L has exactly an eigenvalue 0 of multiplicity two and all the other non-zero eigenvalues have positive real parts. If all the eigenvalues of L are real, which includes the undirected topology as a special case, inequality (14) will be written as 8β > > < α4τ; 2β 4 Reðμi Þ > > τ2 τ þ 40: : α αjμi j2 Then, we have the following corollary. Corollary 2. Suppose that all the non-zero eigenvalues of L are positive, multi-agent system (3) reaches couple-group consensus asymptotically if and only if L has exactly an eigenvalue 0 of multiplicity two and 8 β > > < α4τ; ð17Þ 4 > 2 2β > 40; : f τ; μi ¼ τ τ þ α αμi where μi are the non-zero eigenvalues of L, i ¼ 3; …; n þ m. Remark 5. If all the non-zero eigenvalues of L are positive, it is obvious that inequality (17) will be satisfied for any sufficiently small τ, which is independent of α and β. 5. Simulation examples 5.1. Couple-group consensus of a multi-agent system with undirected topology Consider a multi-agent system (3) with undirected topology, where 3 2 4 3 1 1 1 6 3 3 0 1 1 7 7 6 7 6 6 1 0 0 7 L ¼ 6 1 0 7: 7 6 2 2 5 4 1 1 0 1 1 0 2 2
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Numerical computation shows that L has eigenvalues μ1 ¼ 0; μ2 ¼ 0; μ3 ¼ 1:2984; μ4 ¼ 3 and μ5 ¼ 7:7016. The states of f ðτ; μi Þði ¼ 3; 4; 5Þ vs. τ are shown in Fig. 1, where α ¼ 0:8 and β ¼ 1:0. It can be seen from Fig. 1 that f ðτ; μi Þ40ði ¼ 3; 4Þ for any τ and f ð0:2943; μ5 Þ ¼ 0. From Corollary 2, couple-group consensus will be achieved if we take τ ¼ 0:2. Φ has an eigenvalue 1 of multiplicity four and the rest eigenvalues are 0:35896; 0:81865; 0:770:07746i and 0:8701670:15713i, respectively; whose modulus are less than 1. Fig. 2 demonstrates the position and velocity states of all agents with initial conditions ξð0Þ ¼ ½5; 1; 2; 1; 3T and ζð0Þ ¼ ½0; 0:3; 1; 0:2; 0:3T . Couple-group consensus is achieved as guaranteed by theory. 4
μ3
3.5
μ4 μ5
3 2.5
f(τ,μi)
2 1.5 1 0.5 0 −0.5 −1
0
0.5
1
1.5
τ
2
Fig. 1. The states of f ðτ; μi Þ vs. τ, i ¼ 3; 4; 5. 10
ξ
5 0 −5
0
2
4
6
8
10
6
8
10
time 6 4
ζ
2 0 −2 −4
0
2
4
time Fig. 2. Position and velocity states of all agents, where α ¼ 0:8, β ¼ 1:0 and τ ¼ 0:2.
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20
ξ
10 0 −10 0
5
10
15
20
25
15
20
25
time 100
ζ
50 0 −50 −100
0
5
10
time Fig. 3. Position and velocity states of all agents, where α ¼ 0:8, β ¼ 1:0 and τ ¼ 0:3.
The position and velocity states of all agents are shown in Fig. 3, where α ¼ 0:8, β ¼ 1:0, τ ¼ 0:3 and the initial conditions are the same as which in Fig. 2. Couple-group consensus of the multiagent system cannot be reached as f ð0:3; μ5 Þo0.
5.2. Couple-group consensus of a multi-agent system with directed topology Consider a multi-agent system (3) with directed topology, where 3 2 1 1 0 1 1 6 1 1 0 0 0 7 7 6 7 6 6 0 7 L ¼ 6 0 1 1 0 7: 7 6 0 0 1 1 5 4 0 1 1 0 1 1 Numerical computation shows that L has eigenvalues μ1 ¼ 0; μ2 ¼ 0; μ3 ¼ 1; μ4 ¼ 2 þ i and μ5 ¼ 2i. We take α ¼ 0:8 and β ¼ 1:0 as well. Fig. 4 shows the states of f ðτ; μi Þði ¼ 3; 4Þ vs. τ. f ðτ; μ5 Þ is not shown in Fig. 4 as f ðτ; μ4 Þ ¼ f ðτ; μ5 Þ. From Theorem 1, couple-group consensus will be achieved if τ ¼ 0:5. Φ has an eigenvalue 1 of multiplicity four and the rest eigenvalues are 0:4461370:71411i; 0:5538770:21411i and 0:7570:3708i, respectively; whose modulus are less than 1. Evolutions of the position and velocity states of all agents are shown in Fig. 5, where ξð0Þ ¼ ½1:2; 1; 2; 1; 0T and ζð0Þ ¼ ½0:2; 0:3; 0:1; 1; 1:4T . It can be seen from Fig. 5 that agents 1, 2, 3 and agents 4, 5 reach consensus, respectively. Fig. 6 shows the position and velocity states of all agents, where α ¼ 0:8, β ¼ 1:0, τ ¼ 0:7 and the initial conditions are the same as which in Fig. 5. Couple-group consensus of system (3) cannot be reached as f ð0:7; μ4 Þo0.
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μ3 μ4
4
f(τ,μi)
3
2
1
0
−1 0
0.5
1
1.5
τ
2
Fig. 4. The states of f ðτ; μi Þ vs. τ, i¼ 3,4. 20
ξ
15 10 5 0 −5
0
2
4
6
8
10
time 2
ζ
1 0 −1 0
5
10
15
20
time Fig. 5. Position and velocity states of all agents, where α ¼ 0:8, β ¼ 1:0 and τ ¼ 0:5.
6. Conclusion In this paper, couple-group consensus problem for discrete-time second-order multi-agent systems is investigated for networks with fixed communication topology. Consensus protocol is designed and some necessary and/or sufficient conditions are established to ensure second-order couple-group consensus. It is found that couple-group consensus will be reached only if the nonzero eigenvalues of the Laplacian matrix all have positive real parts. Simulation examples are presented to demonstrate the effectiveness of the theoretical results.
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40
ξ
30 20 10 0 0
5
10
15
20
25
30
20
25
30
time
8 6
ζ
4 2 0 −2 −4 0
5
10
15
time Fig. 6. Position and velocity states of all agents, where α ¼ 0:8, β ¼ 1:0 and τ ¼ 0:7.
There are still a number of issues deserving to be further investigated, such as group consensus of multi-agent systems with switching topologies, relaxing the assumptions on the adjacency matrix, designing new consensus protocols, and so on. Some of them will be discussed in the future. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grants 61074043 and 61174038, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20113219110026, the 333 Project (BRA2011143), and the Qing Lan Project.
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[8] P. Lin, K.Y. Qin, H.M. Zhao, M. Sun, A new approach to average consensus problems with multiple time-delays and jointly-connected topologies, Journal of the Franklin Institute 349 (1) (2012) 293–304. [9] W. Ren, E.M. Aktins, Distributed multi-vehicle coordinated control via local information exchange, International Journal of Robust and Nonlinear Control 17 (10–11) (2007) 1002–1033. [10] M.J. Park, O.M. Kwon, J.H. Park, S.M. Lee, Simplified stability criteria for fuzzy Markovian jumping Hopfield neural networks of neutral type with interval time-varying delays, Expert Systems with Applications 39 (5) (2012) 5625–5633. [11] W.W. Yu, G.R. Chen, M. Cao, Some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems, Automatica 46 (6) (2010) 1089–1095. [12] P. Lin, Y.M. Jia, Further results on decentralised coordination in networks of agents with second-order dynamics, IET Control Theory and Applications 3 (7) (2009) 957–970. [13] W.W. Yu, G.R. Chen, M. Cao, J. Kurths, Second-order consensus for multiagent systems with directed topologies and nonlinear dynamics, IEEE Transactions on Systems, Man and Cybernetics, Part B: Cybernetics 40 (3) (2010) 881–891. [14] W.W. Yu, W.X. Zheng, G.R. Chen, W. Ren, J.D. Cao, Second-order consensus in multi-agent dynamical systems with sampled position data, Automatica 47 (7) (2011) 1496–1503. [15] D.M. Xie, S.K. Wang, Consensus of second-order discrete-time multi-agent systems with fixed topology, Journal of Mathematical Analysis and Applications 387 (1) (2012) 8–16. [16] Y.P. Gao, J.W. Ma, M. Zuo, T.Q. Jiang, J.P. Du, Consensus of discrete-time second-order agents with time-varying topology and time-varying delays, Journal of the Franklin Institute 349 (8) (2012) 2598–2608. [17] J.Y. Yu, L. Wang, Group consensus of multi-agent systems with undirected communication graphs, in: Proceedings of the Asian Control Conference, Hong Kong, China, 2009, pp. 105–110. [18] J.Y. Yu, L. Wang, Group consensus of multi-agent systems with directed information exchange, International Journal of Systems Science 43 (2) (2012) 334–348. [19] J.Y. Yu, L. Wang, Group consensus in multi-agent systems with switching topologies and communication delays, Systems and Control Letters 59 (6) (2010) 340–348. [20] C. Tan, G.P. Liu, G.R. Duan, Couple-group consensus of multi-agent systems with directed and fixed topology, in: Proceedings of the Chinese Control Conference, Yantai, China, 2011, pp. 6515–6520. [21] Y.Z. Feng, S.Y. Xu, B.Y. Zhang, Group consensus control for double integrator dynamic multi-agent systems with fixed communication topology, International Journal of Robust and Nonlinear Control, doi: http://dx.doi.org/ 10.1002/rnc.2904. [22] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, England, 1985. [23] P.C. Parks, V. Hahn, Stability Theory, Prentice Hall, USA, 1993.
Journal of the Franklin Institute 350 (2013) 3277–3292 www.elsevier.com/locate/jfranklin
Couple-group consensus for multi-agent networks of agents with discrete-time second-order dynamics Yuanzhen Fenga,b, Junwei Luc, Shengyuan Xua,n, Yun Zoua a
School of Automation, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, PR China b Department of Basic Science, Nanjing College for Population Program Management, Nanjing 210042, Jiangsu, PR China c School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042, Jiangsu, PR China Received 7 December 2012; received in revised form 1 July 2013; accepted 4 July 2013 Available online 8 August 2013
Abstract This paper considers the couple-group consensus problem for multi-agent networks with fixed and directed communication topology, where all agents are described by discrete-time second-order dynamics. Consensus protocol is designed such that some agents in a network reach a consistent value, while other agents reach another consistent value. The convergence of the system matrix is discussed based on the tools from matrix theory. An algebraic condition is established to guarantee couple-group consensus. Moreover, for a given communication topology, a theorem is derived on how to select proper control parameters and sampling period for couple-group consensus to be reached. Finally, simulation examples are presented to validate the effectiveness of the theoretical results. & 2013 Published by Elsevier Ltd. on behalf of The Franklin Institute
1. Introduction Recent years have witnessed the growing interest in cooperative control of multi-agent systems due to its wide applications in synchronization of coupled oscillators, rendezvous in space, and other areas. In these applications, the states of all agents are required to reach an agreement on some consistent value, which is known as the consensus problem. Up to now, many theoretical results have been achieved on consensus problems for multiagent systems [1–16]. A simple but compelling discrete-time model was proposed in [1], where a n
Corresponding author. Tel./fax: +86 25 84315463. E-mail addresses: [email protected], [email protected] (S. Xu).
0016-0032/$32.00 & 2013 Published by Elsevier Ltd. on behalf of The Franklin Institute http://dx.doi.org/10.1016/j.jfranklin.2013.07.004
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local control law was designed for all agents to move towards the same direction. Simulations were given to show that the headings of all agents will reach an agreement even if the neighbor sets of all agents may change as system evolves. By using the Lyapunov theory, consensus criteria were established in [3], under which multi-agent systems with switching topologies and time delays will reach average consensus asymptotically. In [7], the authors investigated the consensus problem for multi-agent systems with external disturbances and model uncertainty, where a model transformation was introduced and consensus conditions were given for multiagent systems to reach consensus with desired H 1 performance. Consensus protocols were designed and some necessary and/or sufficient conditions were established to guarantee the second-order consensus in [9]. By introducing generalized algebraic connectivity, sufficient conditions were derived in [13] for multi-agent systems with nonlinear dynamics to reach consensus. The second-order consensus problem for discrete-time multi-agent systems was addressed in [15,16], respectively. Note that the above references all consider such consensus where the states of all agents converge to the same consensus value. Due to the changes of situations or cooperative tasks, the consensus values may be different for agents from different sub-networks [17–19]. Group consensus, which includes the aforementioned consensus as a special case, was first introduced in [17] to represent such consensus where the states of all agents in the same sub-network reach the same consistent value while there is no agreement among different sub-networks. Average group consensus problems for multi-agent systems with undirected and balanced topologies were investigated in [17,18], respectively, where all agents have first-order dynamics. By applying double-tree-form transformation and Lyapunov direct method, the work in [17,18] was extended to the case of networks with switching topologies and time delays in [19]. Couple-group consensus of high-order multi-agent systems was addressed in [20], where couple-group consensus was guaranteed by the Hurwitz stability of the coefficient matrix. In this paper, we follow the work of [17,18,21] to study the couple-group consensus problem for multi-agent systems with fixed communication topology. Different from [17–19], all agents considered here have discrete-time second-order dynamics. As a comparison, a necessary and sufficient condition was given in [20] to ensure couple-group consensus of high-order multiagent systems. However, how to select proper control gains still remains unknown for the consensus conditions in [20] to be satisfied. The group consensus problem for multi-agent systems with continuous-time second-order dynamics was considered in [21]. The main contribution of this paper is as follows. Distributed consensus protocol is designed and some necessary and sufficient conditions are established for multi-agent systems to reach couple-group consensus. Furthermore, for a given communication topology, we provide an efficient way on how to select appropriate control parameters and sampling period for couple-group consensus to be achieved, which shows that couple-group consensus will be reached only if the real parts of the non-zero eigenvalues of the Laplacian matrix are positive. The rest of this paper is organized as follows. In Section 2, some preliminaries and the problem formulation are introduced. Couple-group consensus protocol is proposed and the eigenvalues as well as the corresponding eigenvectors of the system matrix are analyzed in Section 3. Main results are given in Section 4. Simulation results are presented in Section 5 to demonstrate the effectiveness of the theoretical results. Section 6 concludes this paper with a discussion. Notation. The following notations will be used throughout the paper. Let In (On) be the n n unit (zero) matrix; Omn be the m n zero matrix and 1n (0n ) be the n-dimensional column vector with all elements being 1 (0). The subscript will be omitted if it is clear from the
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context. ReðλÞ and ImðλÞ denote the real and imaginary part of a complex number λ, respectively. Cnn represents the set of all n n complex matrices. triagfA1 ; A2 ; …; Am g is used to denote the block lower triangular matrix with diagonal blocks A1 to Am. The superscript T means the transpose of a column vector or a matrix. J J refers to the standard Euclidean norm of vectors. j j denotes the absolute value of a real number or the modulus of a complex number. 2. Preliminaries and problem formulation 2.1. Preliminaries In this subsection, we will review some basic concepts in graph theory and introduce some lemmas which will be used in this paper. Let G ¼ ðV; E; AÞ be a directed graph with a finite nonempty set of nodes V ¼ fv1 ; v2 ; …; vn g, a set of edges EDV V and a weighted adjacency matrix A. An edge eij ¼ ðvj ; vi Þ A E means that node vi can receive information from node vj. A ¼ ½aij nn is defined as aij a 0 if eij A E and aij ¼ 0 if eij 2 = E. Moreover, aii ¼ 0 is assumed for all i. A graph is called undirected if aij ¼ aji . The neighbor set of node vi is denoted by N i ¼ fvj jeij A Eg. For a given network system of dynamic agents, a directed graph G will be used to model the information communication among all agents. Lemma 1 (Horn and Johnson [22]). If λ1 ; …; λn are the eigenvalues of AðA A Cnn Þ, the determinant of A is λ1 ⋯λn . Especially, if A is a real matrix, there exists an invertible matrix P such that P1 AP ¼ triagfλ1 ; …; λn g: Lemma 2 (Parks and Hahn [23]). Given a complex coefficient polynomial of order two as follows: hðsÞ ¼ s2 þ ða1 þ ib1 Þs þ a0 þ ib0 ; pffiffiffiffiffiffiffi where i ¼ 1; a1 ; b1 ; a0 and b0 are real constants. Then, h(s) is stable if and only if a1 40 and a1 b1 b0 þ a21 a0 b20 40. 2.2. Problem formulation Suppose that the network system under consideration consists of n+m agents. Each agent is regarded as a node in a directed graph G. To analyze the couple-group consensus problem, without loss of generality, we divide the communication network into two sub-networks, where the first n agents belong to the first sub-network and the remaining m agents belong to the second sub-network. G1 and G2 are used to model the information communication of these two subnetworks. Suppose that each agent is described by the following dynamics: ξi ðk þ 1Þ ¼ ξi ðkÞ þ τζ i ðkÞ; ζ i ðk þ 1Þ ¼ ζ i ðkÞ þ τui ðkÞ;
i ¼ 1; 2; …; n þ m;
ð1Þ
where ξi ðkÞ A R, ζ i ðkÞ A R and ui ðkÞ AR are the position, velocity and control input of agent i at time instant kτ, respectively; τ is the sampling period. Denote ξ1 ðkÞ ¼ ½ξ1 ðkÞ; …; ξn ðkÞT ;
ξ2 ðkÞ ¼ ½ξnþ1 ðkÞ; …; ξnþm ðkÞT ;
ξðkÞ ¼ ½ξ1 ðkÞT ; ξ2 ðkÞT T ;
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ζ 1 ðkÞ ¼ ½ζ 1 ðkÞ; …; ζ n ðkÞT ; ζ 2 ðkÞ ¼ ½ζ nþ1 ðkÞ; …; ζ nþm ðkÞT ; ζðkÞ ¼ ½ζ 1 ðkÞT ; ζ 2 ðkÞT T ; ℓ1 ¼ f1; 2; …; ng; ℓ2 ¼ fn þ 1; n þ 2; …; n þ mg; ℓ ¼ ℓ1 [ ℓ2 ; V 1 ¼ fv1 ; …; vn g; V 2 ¼ fvnþ1 ; …; vnþm g; N i;1 ¼ fvj A V 1 jeij A Eg; N i;2 ¼ fvj A V 2 jeij A Eg: Then, V ¼ V 1 [ V 2 , N i ¼ N i;1 [ N i;2 and Gk ¼ ðV k ; E k ; Ak Þ, where E k ¼ feij ji; j A ℓk g and Ak inherit A, k ¼ 1, 2. Therefore, N i;k can been seen as the neighbor set of agent i in Gk , k¼ 1, 2. Note that E 1 [ E 2 is a subset of E as information transition exists not only among agents in the same sub-network but also among agents from different sub-networks, where E represents the set of edges corresponding to the information communication among all agents in the communication network. Definition 1. Multi-agent system (1) is said to achieve couple-group consensus asymptotically if for any initial conditions, we have lim jξi ðkÞξj ðkÞj ¼ 0
k-1
and
lim jζ i ðkÞζ j ðkÞj ¼ 0
k-1
8 i; jA ℓl ; l ¼ 1; 2:
Assumption 1. We make the following assumptions as in [17,18]: ðA1Þ
j ¼ nþm
∑ aij ¼ 0
j ¼ nþ1
8 iA ℓ1
ðA2Þ
j¼n
∑ aij ¼ 0
j¼1
8 iA ℓ2 :
These assumptions mean that the interaction between the two subgroups is balanced. The problem to be addressed in this paper is to design consensus protocol and establish conditions under which couple-group consensus can be achieved by applying proposed protocol. 3. Consensus protocol and spectrum analysis for the system matrix In this section, consensus protocol will be designed and the spectrum of the system matrix will be analyzed. 3.1. Consensus protocol To solve the couple-group consensus problem for multi-agent system (1), the following consensus protocol: 8 ∑ a ½αðξj ðkÞξi ðkÞÞ þ βðζ j ðkÞζ i ðkÞÞ þ ∑ aij ½αξj ðkÞ þ βζ j ðkÞ 8 iA ℓ1 > > < 8 vj A N i;1 ij 8 vj A N i;2 ui ðkÞ ¼ ∑ aij ½αξj ðkÞ þ βζ j ðkÞ þ ∑ aij ½αðξj ðkÞξi ðkÞÞ þ βðζ j ðkÞζ i ðkÞÞ 8 iA ℓ2 > > : 8 vj A N i;1 8 vj A N i;2 ð2Þ is proposed, where α40 and β40 are the control parameters to be determined; aij Z 0 for all i, jA ℓk ðk ¼ 1; 2Þ and aij A R for all i A ℓ1 , jA ℓ2 or iA ℓ2 , jA ℓ1 . By applying protocol (2), the dynamics of system (1) can be re-written as " # " # ξðk þ 1Þ ξðkÞ ¼Φ ; ð3Þ ζðk þ 1Þ ζðkÞ
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where Φ ¼ I þ τΨ with
" Ψ¼
ð4Þ
Onþm
I nþm
αL
βL
#
and the Laplacian matrix L ¼ ½lij is defined as 8 if ia j a > < ij nþm lij ¼ ∑ aij otherwise i; j ¼ 1; 2; …; n þ m: > : j ¼ 1; jai
ð5Þ
From (4), one gets λi ðΦÞ ¼ 1 þ τλi ðΨ Þ;
i ¼ 1; 2; …; 2ðn þ mÞ;
ð6Þ
where λi ðΦÞ and λi ðΨ Þ are the i-th eigenvalues of Φ and Ψ , respectively; and Φ has the same eigenvectors as Ψ . For the linear system (3), the eigenvalues of matrix Φ play key roles in the convergence analysis. In the following, we put emphasis on the analysis for the eigenvalues and corresponding eigenvectors or generalized eigenvectors of Ψ . Then, by Relations (4) and (6), it is easy to derive the eigenvalues and the eigenvectors of Φ. To proceed with our analysis, we first give a required lemma. Lemma 3 (Yu and Wang [17]). Under Assumption 1, 0 is an eigenvalue of L of geometric multiplicity at least two. Proof. The definition of L in (5) ensures the property that ∑nþm j ¼ 1 lij ¼ 0 for all i ¼ 1; …; n þ m. So, 0 is an eigenvalue of L. Furthermore, one can verify that Lqi ¼ 0 qi ;
i ¼ 1; 2;
where q1 ¼ ½1Tn ; 0Tm T ;
q2 ¼ ½0Tn ; 1Tm T :
ð7Þ
Thus, q1 and q2 are two linearly independent right eigenvectors of L corresponding to zero eigenvalues. This completes the proof. □ 3.2. Eigenvalues and eigenvectors of the system matrix To find out the eigenvalues of Ψ , one needs to solve its characteristic equation " #! λI nþm I nþm detðλI 2ðnþmÞ Ψ Þ ¼ det ¼ 0; αL λI nþm þ βL where detðÞ represents the determinant of a matrix, detðλIΨ Þ is the characteristic polynomial of Ψ . By doing some computations as in [9], we have detðλIΨ Þ ¼ detðλIðλI þ βLÞαLðIÞÞ ¼ detðλ2 I þ ðα þ λβÞLÞ for the fact that I and L are commutable.
ð8Þ
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Suppose that μi are the eigenvalues of L, then λ2 þ ðα þ λβÞμi are the eigenvalues of matrix λ I þ ðα þ λβÞL, i ¼ 1; …; n þ m. Therefore, we know from Lemma 1 that 2
nþm
detðλIΨ Þ ¼ detðλ2 I þ ðα þ λβÞLÞ ¼ ∏ ðλ2 þ ðα þ λβÞμi Þ;
ð9Þ
i¼1
which implies that λ2i;7 ðΨ Þ þ ðα þ λi;7 ðΨ ÞβÞμi ¼ 0; where λi;7 ðΨ Þ are the eigenvalues of Ψ corresponding to μi . Hence, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi βμi 7 β2 μ2i 4αμi λi;7 ðΨ Þ ¼ 2 and μi ¼
λ2i;7 ðΨ Þ α þ βλi;7 ðΨ Þ
ð10Þ
ð11Þ
ð12Þ
can be inferred from Eq. (10). From Eqs. (11) and (12), we know that μi ¼ 0 is equivalent to λi;7 ðΨ Þ ¼ 0. So, we have the following result about the relationship of the zero eigenvalues of L and Ψ . Lemma 4. Ψ has an eigenvalue 0 of multiplicity1 four if and only if L has an eigenvalue 0 of multiplicity two. Remark 1. From Eq. (6), Φ has an eigenvalue 1 of multiplicity four if and only if L has an eigenvalue 0 of multiplicity two. Similar to the analysis in [21], we set out to solve the eigenvectors and generalized eigenvectors of Ψ corresponding to zero eigenvalues. Let w ¼ ½wT1 ; wT2 T be a right eigenvector of Ψ corresponding to zero eigenvalues, where w1 ; w2 A Rnþm . Then, we have Ψ w ¼ 02ðnþmÞ , i.e. # " #" # " Onþm I nþm 0nþm w1 ¼ ; 0nþm αL βL w2 which implies w2 ¼ 0nþm and Lw1 ¼ 0nþm . From Lemma 3, there exist two linearly independent right eigenvectors w1;r and w3;r of Ψ corresponding to zero eigenvalues, where w1;r ¼ ½qT1 ; 0Tnþm T and w3;r ¼ ½qT2 ; 0Tnþm T with q1 ¼ ½1Tn ; 0Tm T and q2 ¼ ½0Tn ; 1Tm T . It can be verified that ðΨ 0 IÞw2;r ¼ w1;r and ðΨ 0 IÞw4;r ¼ w3;r , where w2;r ¼ ½0Tnþm ; qT1 T and w4;r ¼ ½0Tnþm ; qT2 T . Hence, w2;r and w4;r are two generalized right eigenvectors of Ψ corresponding to zero eigenvalues. Suppose that p1 and p2 are left eigenvectors of L corresponding to zero eigenvalues which satisfy pT1 q1 ¼ 1 and pT2 q2 ¼ 1. Similarly, it can be proved that w2;l ¼ ½0Tnþm ; pT1 T and w4;l ¼ ½0Tnþm ; pT2 T are two left eigenvectors of Ψ corresponding to zero eigenvalues. 1
In the sequel, for the sake of simplicity, we simply write algebraic multiplicity as multiplicity.
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w1;l ¼ ½pT1 ; 0Tnþm T and w3;l ¼ ½pT2 ; 0Tnþm T are two generalized left eigenvectors of Ψ corresponding to zero eigenvalues.
4. Main results In this section, necessary and sufficient conditions will be derived to solve the couple-group consensus problem for multi-agent system (3) with fixed communication topology. Theorem 1. By applying consensus protocol (2), multi-agent system (1) achieves couple-group consensus asymptotically if and only if Φ has exactly an eigenvalue 1 of multiplicity four and all the other eigenvalues lie inside the unit circle. Furthermore, we have the following results about the final consensus values: jξi ðkÞpT11 ξ1 ð0ÞpT12 ξ2 ð0ÞkpT11 ζ 1 ð0ÞkpT12 ζ 2 ð0Þj-0
8 iA ℓ1
jξi ðkÞpT21 ξ1 ð0ÞpT22 ξ2 ð0ÞkpT21 ζ 1 ð0ÞkpT22 ζ 2 ð0Þj-0 ζ i ðkÞ-pT11 ζ 1 ð0Þ þ pT12 ζ 2 ð0Þ 8i A ℓ1 ζ i ðkÞ-pT21 ζ 1 ð0Þ þ pT22 ζ 2 ð0Þ 8i A ℓ2
8 iA ℓ2
as k-1, where p1 ¼ ½pT11 ; pT12 T and p2 ¼ ½pT21 ; pT22 T are left eigenvectors of L associated with zero eigenvalues with pT1 q1 ¼ 1 and pT2 q2 ¼ 1 (p11 ; p21 A Rn ; p12 ; p22 A Rm ); q1 and q2 are defined in Eq. (7). Proof (Sufficiency) Let J be the Jordan canonical form of Φ, then there exists an invertible matrix P ¼ ½w1;r ; w2;r ; w3;r ; w4;r ; … such that P1 ΦP ¼ J; where " J¼
J1
O4ð2ðnþmÞ4Þ
Oð2ðnþmÞ4Þ4
J2
2
# ;
1 1 60 1 6 J1 ¼ 6 40 0 0 0
0 0 1 0
3 0 07 7 7; 15 1
J2 represents the Jordan block corresponding to the other eigenvalues of Φ. To facilitate our analysis, we partition P into block form such as P ¼ ½P1 ; P2 ; where P1 ¼ ½w1;r ; w2;r ; w3;r ; w4;r , P2 is the block composed of right eigenvectors or generalized right eigenvectors corresponding to the other eigenvalues of Φ. In a similar way, P1 will be partitioned into " # Q1 P1 ¼ ; Q2
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where
2
wT1;l
3
6 wT 7 6 2;l 7 7 Q1 ¼ 6 6 wT 7; 4 3;l 5 wT4;l Q2 is the block composed of left eigenvectors or generalized left eigenvectors corresponding to the other eigenvalues of Φ. Then, we have Φk ¼ P1 J k1 Q1 þ P2 J k2 Q2 ; where 2
1 60 6 P1 J k1 Q1 ¼ P1 6 40 0
k 0 1 0 0 1 0 0
ð13Þ 2 3 1n pT11 0 6 6 1m pT21 07 7 Q ¼ 7 1 6 6 O k5 4 O 1
1n pT12 1m pT22
k1n pT11 k1m pT21
O
1n pT11
O
1m pT21
3 k1n pT12 7 k1m pT22 7 7 1n pT12 7 5 1m pT22
and J2 satisfies lim J k2 ¼ O2ðnþmÞ4 :
k-1
When the communication topology is fixed, system (3) will evolve according to the following dynamics: " # " # ξðkÞ ξð0Þ k ¼Φ : ζðkÞ ζð0Þ Based on Eq. (13), we obtain that 2 32 1 3 3 2 ξ1 ðkÞ 1n pT11 1n pT12 k1n pT11 k1n pT12 ξ ð0Þ 6 2 7 6 7 T T T T 76 2 6 ξ ðkÞ 7 6 1m p21 1m p22 k1m p21 k1m p22 76 ξ ð0Þ 7 76 7 6 7 lim 6 6 1 76 6 1 7 O 1n pT11 1n pT12 7 k-1 4 ζ ðkÞ 5 4 O 54 ζ ð0Þ 5 2 ζ ðkÞ O O 1m pT21 1m pT22 ζ 2 ð0Þ 2 32 1 3 3 2 1n pT11 1n pT12 k1n pT11 k1n pT12 ξ ð0Þ ξ1 ð0Þ 6 2 7 6 7 7 6 k 6 ξ ð0Þ 7 6 1m pT21 1m pT22 k1m pT21 k1m pT22 76 ξ2 ð0Þ 7 76 7 6 7 ¼ lim Φ6 6 1 76 6 1 7 O 1n pT11 1n pT12 7 k-1 4 ζ ð0Þ 5 4 O 54 ζ ð0Þ 5 O O 1m pT21 1m pT22 ζ 2 ð0Þ ζ 2 ð0Þ 2 1 3 2 3 1n pT11 ξ1 ð0Þ þ 1n pT12 ξ2 ð0Þ þ k1n pT11 ζ1 ð0Þ þ k1n pT12 ζ2 ð0Þ ξ ð0Þ 6 2 7 6 7 T 1 T 2 T 1 T 2 1 p ξ ð0Þ þ 1 p ξ ð0Þ þ k1 p ζ ð0Þ þ k1 p ζ ð0Þ 6 6 7 7 ξ ð0Þ m m m m 21 22 21 22 7 6 7 ¼ 0: ¼ lim P1 J k1 Q1 6 1 T 1 T 2 6 6 7 7 k-1 1n p11 ζ ð0Þ þ 1n p12 ζ ð0Þ 4 ζ ð0Þ 5 4 5 1 2 2 T T 1m p21 ζ ð0Þ þ 1m p22 ζ ð0Þ ζ ð0Þ Therefore, couple-group consensus is achieved asymptotically.
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(Necessity) We know from Lemma 3 and Remark 1 that Φ has an eigenvalue 1 of multiplicity at least four. If the sufficient condition does not hold, Φ has at least five eigenvalues whose modulus are greater than or equal to 1. Thus, rank ðJ k Þ44 holds as k-1. From the proof of the sufficiency, it is obvious that couple-group consensus is achieved if and only if 2 3 1n pT 6 7 6 1m qT 7 7 lim Φk -6 6 1n wT 7; k-1 4 5 1m z T where p; q; w and z are arbitrary column vectors with appropriate dimensions. This implies that rank ðΦk Þr 4 as k-1. Note the fact that rank ðΦk Þ ¼ rank ðJ k Þ, a contrary is resulted in. □ Remark 2. The final consensus values are specified in this paper when couple-group consensus is achieved, which were not discussed in [20]. Remark 3. If the network under consideration is divided into multi sub-networks, consensus protocol (2) and algebraic condition in Theorem 1 can be extended for multi-group consensus to be achieved. We notice that the algebraic condition in Theorem 1 is not straightforward to be checked. Now, for a given communication topology, the following theorem is proposed for choosing appropriate control parameters and sampling period to ensure couple-group consensus. Theorem 2. By applying consensus protocol (2), multi-agent system (1) reaches couple-group consensus asymptotically if and only if L has exactly an eigenvalue 0 of multiplicity two and 8 β > > > 4τ; < α ð14Þ 4 Reðμi Þ 4 Im2 ðμi Þ > 2 2 2β > f τ; μ τ þ τ 40; ¼ ðβατÞ > i : α αjμi j2 jμi j4 where μi are the non-zero eigenvalues of L, i ¼ 3; …; n þ m. Before giving the proof for Theorem 2, we present the following preliminaries. From Lemma 1, L is similar to triagfμ1 ; μ2 ; …; μnþm g, where μ1 ¼ μ2 ¼ 0. Then, Ψ is similar to " # I nþm Onþm : αtriagfμ1 ; μ2 ; …; μnþm g βtriagfμ1 ; μ2 ; …; μnþm g By using some elementary similarity transformations, we know that Ψ is similar to (" # " #) 0 1 0 1 ; …; : triag αμnþm βμnþm αμ1 βμ1 Therefore, Φ is similar to (" # " 1 1 τ ; …; triag ατμ ατμ1 1βτμ1 nþm
τ 1βτμnþm
#) :
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Consider " B¼
1
τ
#
ατλ 1βτλ
ðjλj40Þ;
its characteristic polynomial is hðs; λÞ ¼ detðsIBÞ ¼ ðs1Þ2 þ βτλðs1Þ þ ατ2 λ: By applying the bilinear transformation s ¼ ðs þ 1Þ=ðs1Þ, we get sþ1 2 ; λ ¼ αðτjλjÞ2 ðs1Þ2 þ 2βτλj2 ðs1Þ þ 4λ: λðs1Þ h s1 Define
sþ1 ;λ λðs1Þ h β 2β 4 ReðλÞ 4 ImðλÞ s1 2 1 s þ 1 þ gðs; λÞ ¼ ¼ s þ 2 i ; 2 ατ ατ αðτjλjÞ2 αðτjλjÞ2 αðτjλjÞ 2
ð15Þ then, by Lemma 2, gðs; λÞ is stable if and only if 8 β > > 1 40; 2 > < ατ 2 β 2β 4 ReðλÞ 4 ImðλÞ 2 > > >4 1 1 þ 40: : ατ ατ αðτjλjÞ2 αðτjλjÞ2
ð16Þ
Now, we give the proof for Theorem 2. Proof. From Theorem 1, couple-group consensus will be achieved if and only if Φ has exactly an eigenvalue 1 of multiplicity four and 1 is the unique eigenvalue of maximum modulus. By Lemma 4 and above analysis, couple-group consensus will be achieved if and only if L has exactly an eigenvalue 0 of multiplicity two and hðs; μi Þ are Schur stable, i ¼ 3; …; n þ m. Note that hðs; λÞ is Schur stable if and only if gðs; λÞ is stable. From inequality (16), gðs; μi Þ are stable if and only if 8 β > > > 1 40 2 > < ατ 2 β 2β 4 Reðμi Þ 4 Imðμi Þ 2 > > > 4 1 þ 1 40 > : ατ ατ αðτjμi jÞ2 αðτjμi jÞ2 hold for i ¼ 3; …; n þ m. This completes the proof.
□
Remark 4. Compared with [15], the sampling period is considered in this paper, which increases the difficulty in choosing control parameters for couple-group consensus to be reached. For a given communication topology, one may first fix the values of α and β, then try to find some proper sampling period for inequality (14) to be satisfied.
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Note that inequality (14) is satisfied only if β 4τ α and τ2
2β 4 Reðμi Þ τþ 40; α αjμi j2
which imply that all the non-zero eigenvalues of the Laplacian matrix have positive real parts. Corollary 1. By applying consensus protocol (2), multi-agent system (1) reaches couple-group consensus asymptotically only if L has exactly an eigenvalue 0 of multiplicity two and all the other non-zero eigenvalues have positive real parts. If all the eigenvalues of L are real, which includes the undirected topology as a special case, inequality (14) will be written as 8β > > < α4τ; 2β 4 Reðμi Þ > > τ2 τ þ 40: : α αjμi j2 Then, we have the following corollary. Corollary 2. Suppose that all the non-zero eigenvalues of L are positive, multi-agent system (3) reaches couple-group consensus asymptotically if and only if L has exactly an eigenvalue 0 of multiplicity two and 8 β > > < α4τ; ð17Þ 4 > 2 2β > 40; : f τ; μi ¼ τ τ þ α αμi where μi are the non-zero eigenvalues of L, i ¼ 3; …; n þ m. Remark 5. If all the non-zero eigenvalues of L are positive, it is obvious that inequality (17) will be satisfied for any sufficiently small τ, which is independent of α and β. 5. Simulation examples 5.1. Couple-group consensus of a multi-agent system with undirected topology Consider a multi-agent system (3) with undirected topology, where 3 2 4 3 1 1 1 6 3 3 0 1 1 7 7 6 7 6 6 1 0 0 7 L ¼ 6 1 0 7: 7 6 2 2 5 4 1 1 0 1 1 0 2 2
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Numerical computation shows that L has eigenvalues μ1 ¼ 0; μ2 ¼ 0; μ3 ¼ 1:2984; μ4 ¼ 3 and μ5 ¼ 7:7016. The states of f ðτ; μi Þði ¼ 3; 4; 5Þ vs. τ are shown in Fig. 1, where α ¼ 0:8 and β ¼ 1:0. It can be seen from Fig. 1 that f ðτ; μi Þ40ði ¼ 3; 4Þ for any τ and f ð0:2943; μ5 Þ ¼ 0. From Corollary 2, couple-group consensus will be achieved if we take τ ¼ 0:2. Φ has an eigenvalue 1 of multiplicity four and the rest eigenvalues are 0:35896; 0:81865; 0:770:07746i and 0:8701670:15713i, respectively; whose modulus are less than 1. Fig. 2 demonstrates the position and velocity states of all agents with initial conditions ξð0Þ ¼ ½5; 1; 2; 1; 3T and ζð0Þ ¼ ½0; 0:3; 1; 0:2; 0:3T . Couple-group consensus is achieved as guaranteed by theory. 4
μ3
3.5
μ4 μ5
3 2.5
f(τ,μi)
2 1.5 1 0.5 0 −0.5 −1
0
0.5
1
1.5
τ
2
Fig. 1. The states of f ðτ; μi Þ vs. τ, i ¼ 3; 4; 5. 10
ξ
5 0 −5
0
2
4
6
8
10
6
8
10
time 6 4
ζ
2 0 −2 −4
0
2
4
time Fig. 2. Position and velocity states of all agents, where α ¼ 0:8, β ¼ 1:0 and τ ¼ 0:2.
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20
ξ
10 0 −10 0
5
10
15
20
25
15
20
25
time 100
ζ
50 0 −50 −100
0
5
10
time Fig. 3. Position and velocity states of all agents, where α ¼ 0:8, β ¼ 1:0 and τ ¼ 0:3.
The position and velocity states of all agents are shown in Fig. 3, where α ¼ 0:8, β ¼ 1:0, τ ¼ 0:3 and the initial conditions are the same as which in Fig. 2. Couple-group consensus of the multiagent system cannot be reached as f ð0:3; μ5 Þo0.
5.2. Couple-group consensus of a multi-agent system with directed topology Consider a multi-agent system (3) with directed topology, where 3 2 1 1 0 1 1 6 1 1 0 0 0 7 7 6 7 6 6 0 7 L ¼ 6 0 1 1 0 7: 7 6 0 0 1 1 5 4 0 1 1 0 1 1 Numerical computation shows that L has eigenvalues μ1 ¼ 0; μ2 ¼ 0; μ3 ¼ 1; μ4 ¼ 2 þ i and μ5 ¼ 2i. We take α ¼ 0:8 and β ¼ 1:0 as well. Fig. 4 shows the states of f ðτ; μi Þði ¼ 3; 4Þ vs. τ. f ðτ; μ5 Þ is not shown in Fig. 4 as f ðτ; μ4 Þ ¼ f ðτ; μ5 Þ. From Theorem 1, couple-group consensus will be achieved if τ ¼ 0:5. Φ has an eigenvalue 1 of multiplicity four and the rest eigenvalues are 0:4461370:71411i; 0:5538770:21411i and 0:7570:3708i, respectively; whose modulus are less than 1. Evolutions of the position and velocity states of all agents are shown in Fig. 5, where ξð0Þ ¼ ½1:2; 1; 2; 1; 0T and ζð0Þ ¼ ½0:2; 0:3; 0:1; 1; 1:4T . It can be seen from Fig. 5 that agents 1, 2, 3 and agents 4, 5 reach consensus, respectively. Fig. 6 shows the position and velocity states of all agents, where α ¼ 0:8, β ¼ 1:0, τ ¼ 0:7 and the initial conditions are the same as which in Fig. 5. Couple-group consensus of system (3) cannot be reached as f ð0:7; μ4 Þo0.
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μ3 μ4
4
f(τ,μi)
3
2
1
0
−1 0
0.5
1
1.5
τ
2
Fig. 4. The states of f ðτ; μi Þ vs. τ, i¼ 3,4. 20
ξ
15 10 5 0 −5
0
2
4
6
8
10
time 2
ζ
1 0 −1 0
5
10
15
20
time Fig. 5. Position and velocity states of all agents, where α ¼ 0:8, β ¼ 1:0 and τ ¼ 0:5.
6. Conclusion In this paper, couple-group consensus problem for discrete-time second-order multi-agent systems is investigated for networks with fixed communication topology. Consensus protocol is designed and some necessary and/or sufficient conditions are established to ensure second-order couple-group consensus. It is found that couple-group consensus will be reached only if the nonzero eigenvalues of the Laplacian matrix all have positive real parts. Simulation examples are presented to demonstrate the effectiveness of the theoretical results.
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40
ξ
30 20 10 0 0
5
10
15
20
25
30
20
25
30
time
8 6
ζ
4 2 0 −2 −4 0
5
10
15
time Fig. 6. Position and velocity states of all agents, where α ¼ 0:8, β ¼ 1:0 and τ ¼ 0:7.
There are still a number of issues deserving to be further investigated, such as group consensus of multi-agent systems with switching topologies, relaxing the assumptions on the adjacency matrix, designing new consensus protocols, and so on. Some of them will be discussed in the future. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grants 61074043 and 61174038, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20113219110026, the 333 Project (BRA2011143), and the Qing Lan Project.
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