Distributed consensus control for linear multi-agent systems with ...
International Journal of Control Vol. 86, No. 1, January 2013, 95–106
Distributed consensus control for linear multi-agent systems with discontinuous observations Guanghui Wena, Zhongkui Lib, Zhisheng Duana* and Guanrong Chenac a
Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China; b School of Automation, Beijing Institute of Technology, Beijing 100081, China; cDepartment of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China
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(Received 25 May 2012; final version received 5 August 2012) This article addresses the distributed consensus problem of linear multi-agent systems with discontinuous observations over a time-invariant undirected communication topology. Under the assumption that each agent can only intermittently share its outputs with the neighbours, a class of distributed observer-type of protocols are designed and utilised to achieve consensus. By using appropriate matrix decomposition, it is shown that consensus in the closed-loop multi-agent systems under a connected topology can be converted to the simultaneous asymptotic stability of a set of switching systems whose dimensions are the same as each agent. From a multiple Lyapunov functions approach, it is proved that there exists a protocol to guarantee consensus if the communication time rate is larger than a threshold value. Furthermore, a distributed pinning control method is employed to solve the consensus problem on an arbitrary given topology which needs not be connected. Particularly, the questions of what kind of agents and at least how many agents should be pinned are addressed. The effectiveness of the analytical results is finally verified by numerical simulations. Keywords: multi-agent system; consensus; pinning control; discontinuous observation; lyapunov function
1. Introduction Recently, distributed cooperative control of multiagent systems has received considerable attention from various scientific communities due to the growing interest in understanding the intriguing animal group behaviours, such as swarming (Gazi and Passino 2003) and flocking (Su, Wang, and Lin 2009; Wen, Duan, Li, and Chen 2012b), and their potential applications in formation control of satellites (Smith and Hadaegh 2005), teaming of multi-robotics (Ren and Sorensen 2008), and design of sensor networks (Yu, Chen, Wang, and Yang 2009b). Among the numerous research topics in distributed control of multi-agent systems, consensus problem is of particular importance, which refers to designing an appropriate protocol based only on the local relative information among neighbouring agents to guide all agents to reach an agreement (see survey papers Olfati-Saber, Fax, and Murray (2007), Ren, Beard, and Atkins (2007a) and the references therein). Vicsek, Cziro´k, Ben-Jacob, Cohen, and Shochet (1995) introduced a simple yet effective discrete-time model for phase transition of a group of autonomous agents and numerically investigated the angle
*Corresponding author. Email: [email protected] ISSN 0020–7179 print/ISSN 1366–5820 online ß 2013 Taylor & Francis http://dx.doi.org/10.1080/00207179.2012.719637 http://www.tandfonline.com
consensus behaviour of the multi-agent system. By using the tools from the algebraic graph theory, a theoretical explanation of the consensus behavior observed in Vicsek et al. (1995) was first given in Jadbabaie, Lin, and Morse (2003). A general framework of the consensus problem for a network of agents with single-integrator dynamics under a fixed or a switching topology and possible communication time delays was suggested and studied in Olfati-Saber and Murray (2004). The consensus conditions derived in Olfati-Saber and Murray (2004) were further relaxed in Ren and Beard (2005) by proving that consensus in multi-agent systems with single-integrator dynamics can be achieved if and only if the time-varying network topology contains a directed spanning tree frequently enough as the network evolves over time. Meanwhile, the consensus problem for multi-agent systems with second- and higher-order dynamics were addressed (Ren and Atkins 2007; Xie and Wang 2007; Hong, Chen, and Bushnell 2008; Wen, Duan, Yu, and Chen 2012d; Ren, Moore, and Chen 2007b; Zhang and Lewis 2012). Note that most of the above-mentioned works are concerned with the case where the agents are governed by integrator-type dynamics.
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However, multi-agent systems with general linear node dynamics are more popular (Ma and Zhang 2010; Li, Duan, Chen, and Huang 2010; Zhang, Lewis, and Das 2011; Li, Duan, and Chen 2011), which include networks of agents with integrator-type of dynamics as special cases. In Ma and Zhang (2010), from a static output approach, some necessary and sufficient conditions were derived for consensus of multi-agent systems with general linear node dynamics under a fixed directed topology. Consensus in multi-agent systems with general linear node dynamics under a fixed directed topology was investigated with observertype protocols appropriately designed in Li et al. (2010), Zhang et al. (2011), Li et al. (2011). Most of the above-mentioned results on the consensus problem in multi-agent systems with general linear dynamics are obtained based on the assumption that information is transmitted continuously among the agents, i.e. each agent has to share its state or output information with its neighbours all the time. However, this may not always be the case in reality. Sometimes, mobile agents can only communicate with their neighbours over some disconnected time intervals due to, for instance, temporary sonar equipment failures or the presence of communication obstacles, even if the distances among them are less than the communication radius. Yet, equipment failures may be recovered through repairing and communication obstacles may be bypassed as the system evolves in time. Motivated by these facts and based on the works reported in Wen, Duan, and Chen (2012a), Wen, Duan, Li, and Chen (2012c), the consensus problem for multi-agent systems with general linear node dynamics based on intermittent observations is studied in this article, where the ideal assumption that agents could transmit their output information to their neighbours at all times is removed. For convenience of analysis, the communication topology among the agents is assumed to be undirected. For achieving consensus, a new class of consensus protocols are proposed and analysed. By using tools from switching systems theory, it is theoretically shown that consensus in a closed-loop multi-agent system with a connected topology can be ensured if the communication rate is larger than a threshold value. The analytical expression of the threshold value is also explicitly given. By using a distributed pinning-based control method, the results are then extended to consensus in multi-agent systems with an arbitrary topology which needs not be connected. In particular, the questions of what kind of agents and at least how many agents should be pinned for achieving consensus are addressed and answered. Numerical examples are finally given to verify the theoretical analysis.
The rest of this article is organised as follows. Some preliminaries and the model formulation are presented in Section 2. Consensus problem for multi-agent systems with discontinuous observations under a time-invariant connected topology is studied in Section 3. Some extensions are given in Section 4. In Section 5, several numerical simulations are provided for illustration. Conclusions are finally drawn in Section 6. Throughout this article, let N and R be the set of natural and real numbers, respectively, and Rnn be the sets of n n real matrices. Let In (On) be the n n identity (zero) matrices, and 1n (0n) be the n-dimensional column vector with all entries equal to one (zero). Matrices, if not explicitly stated, are assumed to have compatible dimensions. The matrix inequality A 4 B means that both A and B are square symmetric matrices and that A B is positive-definite. diag{a1, a2, . . . , aN} represents a diagonal matrix with ai, i ¼ 1, 2, . . . , N, being its diagonal elements. Notations and k k represent the Kronecker product and the Euclidian norm, respectively.
2. Preliminaries and formulation of the model In this section, some preliminaries and the model formulation for consensus in multi-agent systems with discontinuous observations are introduced.
2.1 Preliminaries An undirected graph G is a pair of (V, E), where V ¼ {1, 2, . . . , N} is a node set and E {(i, j ), i, j 2 V} is an edge set in which an edge is represented by an unordered pair of distinct nodes. Two nodes i, j are adjacent or neighboring, if (i, j) is an edge of graph G, i.e. (i, j) 2 E. A path on G from node i1 to node is is a sequence of ordered edges of the form (ik, ikþ1), k ¼ 1, 2, . . . , s 1. An undirected graph is connected if there exists a path between every pair of distinct nodes, otherwise is disconnected. Only simple graphs are considered here, i.e. multiple edges and self-loops are forbidden in G. A connected subgraph of G, which is maximal, is called a component of G. The adjacency matrix A ¼ [aij]NN of a graph G is defined by aii ¼ 0 for i ¼ 1, 2, . . . , N, and aij ¼ aji 4 0 for (i, j) 2 E but 0 otherwise. The Laplacian matrix is defined as lij ¼ aij, i 6¼ j, and L ¼ [lP ij]NN a lii ¼ N k¼1 ik for i ¼ 1, 2, . . . , N. For an undirected graph G, both its adjacency matrix and Laplacian matrix are symmetric. The following lemmas can be found in graph theory textbooks (e.g. Godsil and Royle 2001).
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International Journal of Control Lemma 1: Suppose that an undirected graph G has m components. Then, there exist a permutation matrix W of order N, such that 2 3 e O L11 O 6 7 L22 O 7 6 O e 7, ð1Þ WT LW ¼ 6 .. .. 6 .. 7 4 . . O 5 . O O e Lmm where e L11 2 Rq1 q1 , e L22 2 Rq2 q2 , . . . , e Lmm 2 Rqm qm are Pm symmetric matrices with zero row sums, with j¼1 qj ¼ N and 1 q1 qm N. Furthermore, R(L) ¼ N m, where R(L) represents the rank of L.
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Remark 1: Any undirected graph G contains at least one component and at most N components. Thus, according to Lemma 1, one has 0 R(L) N 1. Lemma 2: Consider a connected undirected graph G. Then, 0 is a simple eigenvalue of its Laplacian matrix L and all the other eigenvalues of L are positive real numbers. 2.2 Formulation of the model Consider a network of identical agents with linear or linearised dynamics, described by x_ i ðtÞ ¼ Axi ðtÞ þ Bui ðtÞ, ð2Þ yi ðtÞ ¼ Cxi ðtÞ, i ¼ 1, 2, . . . , N, where xi(t) 2 Rn is the state, ui(t) 2 Rm is the control input, yi(t) 2 Rp is the measured output, and A, B and C are constant real matrices with compatible dimensions. The communication topology among the N agents is represented by an undirected graph G consisting of the node set V ¼ {1, 2, . . . , N} and the edge set E {(i, j), i, j 2 V}. An existing edge (i, j) (i 6¼ j) means that agents i and j can obtain information from each other. In some real situations, agents may only sense the outputs of their neighbours over some disconnected time intervals due to the unreliability of communication channels, failure of physical devices, etc. Motivated by this observation, the following distributed observer-type of consensus protocol with discontinuous dynamic output measurements is proposed for each agent i: v_i ðtÞ ¼ Avi ðtÞ þ Bui ðtÞ þ cF ðyi ðtÞ yj ðtÞÞ , ui ðtÞ ¼ Kvi ðtÞ, v_i ðtÞ ¼ Avi ðtÞ, ui ðtÞ ¼ 0,
N X
aij Cðvi ðtÞ vj ðtÞÞ
where vi(t) 2 Rn is the state of the observer embedded in agent i, c 4 0 is the coupling strength, F 2 Rnp and K 2 Rmn are feedback matrices, A ¼ [aij]NN is adjacency matrix of graph G, and ! 4 4 0. Let i(t) ¼ (xi(t)T, vi(t)T)T. Then, it follows from (2) and (3) that _i ðtÞ ¼ A1 i ðtÞ þ c
N P
lij Hj ðtÞ,
t 2 ½k!, k! þ Þ,
j¼1
_i ðtÞ ¼ A2 i ðtÞ,
t 2 ½k! þ , ðk þ 1Þ!Þ,
ð4Þ
k 2 N,
where
, O AþBK O O H¼ , FC FC
A1 ¼
A
BK
A2 ¼
A
O
O
A
,
and L ¼ [lij]NN is the Laplacian matrix of graph G. Remark 2: The consensus problem for multi-agent systems with general linear node dynamics based on intermittent relative state information has been studied recently in Wen et al. (2012a), using the state information of agents which are hard or impossible to obtain in practice. In contrast, the present protocol (3) depends only on the relative output information of neighboring agents. Definition 1: The consensus problem of multi-agent system (2) is solved by protocol (3) if, for any initial conditions, the states of system (4) satisfy lim i ðtÞ j ðtÞ ¼ 0,
t!1
8 i, j ¼ 1, 2, . . . , N:
ð5Þ
Remark 3: From Definition 1, consensus in multiagent system (2) is solved by protocol (3) if and only if both the states of agents and the protocols embedded in agents asymptotically approach the same values, respectively. Finally, the following lemmas (referring to Boyd, Ghaoui, Ferion, and Balakrishnan 1994) are introduced. Lemma 3: The maximum real parts of eigenvalues of a matrix A 2 Rnn is less than , where 4 0, if and only if there exists a matrix P ¼ PT 4 0 such that AT P þ PA þ 2P 5 0:
j¼1
Lemma 4: For matrices A, B, C, and D of appropriate dimensions, one has
t 2 ½k!, k! þ Þ,
t 2 ½k! þ , ðk þ 1Þ!Þ,
k 2 N,
ð3Þ
(1) (A) B ¼ A (B) ¼ (A B), 8 2 R; (2) (A þ B) C ¼ A C þ B C; (3) (A B)(C D) ¼ (AC) (BD).
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3. Main results In this section, the mains results of this article are described and proved.
3.1 Consensus in multi-agent systems under a connected topology In this subsection, distributed consensus in multi-agent system (2) under an undirected connected topology is addressed.
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Assumption 1: The undirected communication topology G is connected. Under Assumption 1, it follows from Lemma 2 that ¼ (1/N )1N is the left eigenvector of the Laplacian matrix L associated with the zero eigenvalue. Let s(t) (s1(t)T, s2(t)T, . . . , sN(t)T)T, where si ðtÞ ¼ i ðtÞ P¼ N 1 j¼1 j ðtÞ, i ¼ 1, 2, . . . , N. Then, N sðtÞ ¼ IN 1N T In ðtÞ, ð6Þ where (t) ¼ (1(t)T, 2(t)T, . . . , N(t)T)T. It is easy to verify that s(t) ¼ 0 if and only if 1(t) ¼ 2(t) ¼ ¼ N(t), for all t 0. It then follows from (3) and (6) that s_ðtÞ ¼ ðIN A1 þ cL HÞsðtÞ, t 2 ½k!, k! þ Þ, s_ðtÞ ¼ ðIN A2 ÞsðtÞ, t 2 ½k! þ , ðk þ 1Þ!Þ, ð7Þ where
A BK , A1 ¼ O AþBK
O O H¼ , FC FC
A2 ¼
A O
O , A
and L ¼ [lij]NN is the Laplacian matrix of graph G. Let Y1 2 RN(N1), Y2 2 R(N1)N, T 2 RNN and a diagonal matrix D 2 R(N1)(N1) be such that T
1 T ¼ ð1N , Y1 Þ, T ¼ , Y2 ! 0 0TN1 T1 LT ¼ ¼ , ð8Þ 0N1 D where the diagonal entries of D are the non-zero eigenvalues of Laplacian matrix L. By using the linear transformation ð9Þ ðtÞ ¼ T1 I2n sðtÞ with (t) ¼ (1(t)T, 2(t)T, . . . , N(t)T)T, it follows from Lemma 4 and (7) that _ ¼ ðIN A1 þ c HÞðtÞ, t 2 ½k!, k! þ Þ, ðtÞ _ ¼ ðIN A2 ÞðtÞ, t 2 ½k! þ , ðk þ 1Þ!Þ, k 2 N, ðtÞ ð10Þ
where A1, A2, H are defined in (7) and is given in (8). Note that 1(t) 0, for all t 0. Thus, the consensus problem for multi-agent system (2) is solved by protocol (3) if and only if i(t), i ¼ 2, 3, . . . , N, converge asymptotically to zero, which is in turn equivalent to that the following N 1 systems z_i ðtÞ ¼ ðIN A1 þ ci HÞzi ðtÞ, t 2 ½k!, k! þ Þ, z_i ðtÞ ¼ ðIN A2 Þzi ðtÞ, t 2 ½k! þ , ðk þ 1Þ!Þ, k 2 N, ð11Þ are simultaneously asymptotically stable, where i, i ¼ 2, 3, . . . , N, are the non-zero eigenvalues of the Laplacian matrix L. Taking linear transformation
I I zi ðtÞ, i ¼ 2, 3, . . . , N, ð12Þ i ðtÞ ¼ O I and using (11), one obtains _ i ðtÞ ¼ Ti i ðtÞ, _ i ðtÞ ¼ A2 i ðtÞ,
t 2 ½k!, k! þ Þ, t 2 ½k! þ , ðk þ 1Þ!Þ,
k 2 N, ð13Þ
where
A þ ci FC O Ti ¼ , ci FC A þ BK
A A2 ¼ O
O , A
and i, i ¼ 2, 3, . . . , N, are the non-zero eigenvalues of the Laplacian matrix L. From the above analysis, one has converted the consensus problem for multi-agent system (2) with protocol (3) to the simultaneously asymptotical stability problem of N 1 switching systems given by (13), which have the same low dimension as a single agent. One can see that the effects of the communication topology on the consensus are characterised by the non-zero eigenvalues of the Laplacian matrix L. Next, an algorithm is presented to construct protocol (3). Algorithm 1: Under Assumption 1 and the condition that (A, B, C) is both stabilisable and detectable, the consensus protocol (3) can be designed as follows. (1) Choose 0 4 0, and select feedback gain matrix K by Ackermann’s formula, such that the real parts of the poles of A þ BK are smaller than 0. (2) Solve the following linear matrix inequality (LMI): AT Q þ QA 2CT C þ 2 0 Q 5 0,
ð14Þ
to obtain one solution Q 4 0. Then, let the feedback gain matrix be F ¼ Q1CT.
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International Journal of Control (3) Take the coupling strength c 1/2, where 2 is the second smallest eigenvalue of the Laplacian matrix L. Lemma 5: The maximum real parts of eigenvalues of matrices Ti are less than 0, where Ti ¼ 1
A þ ci FC
O
ci FC
A þ BK
,
T
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F ¼ Q C , Q is a solution of LMI (14), and i, i ¼ 2, 3, . . . , N, are the non-zero eigenvalues of L. Proof: Similarly to the proof of Proposition 1 in Li et al. (2011), this lemma can be proved. For the readers’ convenience, a sketched proof is given as follows. Let Q 4 0 be a solution of LMI (14). Take F ¼ Q1CT. Then, one gets ðA þ ci FCÞQ1 þ Q1 ðA þ ci FCÞT þ 2 0 Q1 ¼ AQ1 þ Q1 AT 2ci Q1 CT CQ1 þ 2 0 Q1
Pn Proof: With 0 ¼ maxj¼1,2,..., n i¼1 aij , the lemma can be proved by using the Gersˇ gorin disc theorem (Huang 1984). The following theorem is one main result of this articler. Theorem 1: For the multi-agent system (2), under Assumption 1 and moreover (A, B, C) is both stabilisable and detectable, the protocol (3) constructed by Algorithm 1 solves the consensus problem if the communication time rate /! 4 /( þ ) þ ln /[( þ )!], where 0 5 5 0, ¼ maxi¼2,3, . . . , N {max(Wi)/min(P), max(P)/min(Wi)}, 4 0, and matrices Wi and P are the positive-definite solutions of LMIs (17) and (18), respectively. Proof: Construct the following multiple Lyapunov function candidate for the ith switching system of (13): ( i ðtÞT Wi i ðtÞ, t 2 ½k!, k! þ Þ, VðtÞ ¼ i ðtÞT P i ðtÞ, t 2 ½k! þ , ðk þ 1Þ!Þ,
AQ1 þ Q1 AT 2Q1 CT CQ1 þ 2 0 Q1 ,
ð19Þ ð15Þ
where the last inequality of (15) is derived by using the fact c 1/2, with 2 being the second smallest eigenvalue of the Laplacian matrix L. Pre- and post-multiplying (15) by Q and its transpose, according to (14), yields ðA þ ci FCÞQ1 þ Q1 ðA þ ci FCÞT þ 2 0 Q1 5 0: ð16Þ It thus follows from Lemma 3 and (16) that the maximum real parts of the eigenvalues of matrices Ti are less than 0, where A þ ci FC O Ti ¼ : ci FC A þ BK Lemma 6: There exist some Wi 4 0 such that Ti T Wi þ Wi Ti þ 2 Wi 5 0,
Ti ¼
ð17Þ
A þ ci FC
O
ci FC
A þ BK
,
k! þ Þ,
ð20Þ
ðk þ 1Þ!Þ:
ð21Þ
where is defined in Lemma 6. Similarly, one has _ 5 2VðtÞ, VðtÞ
t 2 ½k! þ ,
Note that the switching systems (13) switches at t ¼ k! and t ¼ k! þ , k 2 N. Based on the above analysis, one obtains
5 2 e2 þ2ð!Þ Vð0Þ
Vðk!Þ 5 Vð0Þek :
It follows directly from Lemmas 3 and 5.
Lemma 7: There exists a 0 4 0, such that for all 4 0, the following LMI
O
t 2 ½k!,
where ¼ 2 2(! ) 2ln . From the condition /! 4 /( þ ) þ ln /[( þ )!], one has 4 0. By recursion, for any positive integer k, one gets
i ¼ 2, 3, . . . , N.
where A4 ¼
_ 5 2 VðtÞ, VðtÞ
¼ e Vð0Þ,
A
For t 2 [k!, k! þ ) and an arbitrarily given k 2 N, taking the time derivative of V(t) along the trajectories of system (13) gives
Vð!Þ 5 e2ð!Þ VðÞ
where 5 0,
Proof:
where the positive-definite matrices Wi and P are the solutions of (17) and (18), respectively, k 2 N, i ¼ 2, 3, . . . , N.
AT4 P þ PA4 2P 5 0, O A , has a solution P 4 0.
ð18Þ
ð22Þ
For any t 4 0, there exists an r 2 N such that r! t 5 (r þ 1)!. Then, one has VðtÞ 5 Vðr!Þe2! 5 Vð0Þer þ2! 5 e2! Vð0Þe½ =ð!þ1Þ t ,
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i.e. VðtÞ 5 0 e t ,
for all t 0,
where 0 ¼ e2!V(0), and ¼ /(! þ 1). This indicates that the states of agents exponentially converge to the same.
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Remark 4: In Theorem 1, it is assumed that the agents can obtain the intermittent relative outputs of its neighbours periodically. However, one may extend the results to the consensus in linear multi-agent systems with aperiodically intermittent output communications by using the present approach. Similarly to the proof of Theorem 1, some corresponding theoretical results can be derived, which are omitted here for brevity. Remark 5: To successfully construct protocol (3) according to Algorithm 1, it should be firstly shown that both steps (1) and (2) of Algorithm 1 are feasible for some 0 4 0. Note that there always exist some positive scalar 0 such that steps (1) and (2) are feasible if (A, B, C) is stabilisable and detectable. It is also worth noting that both steps (1) and (2) are feasible for any given 0 4 0 if the matrix triple (A, B, C) is controllable and observable. Remark 6: In the context of multi-agent systems with intermittent observations, one interesting and important issue is what the minimum admissible communication rate is for achieving consensus for a given topology G with fixed parameters and . However, LMIs (17) and (18) are solved independently, which may introduce conservativeness in determining the admissible communication rate to satisfy the consensus condition. From the proof of Theorem 1, one can see that the minimum admissible communication rate, under a given communication topology with fixed parameters and , can be obtained by minimising in Theorem 1. Actually, the minimum can be obtained by solving the following optimisation problem: (1) Minimise i subject to: Wi 4 0, P 4 Wi, P 5 iWi, Ti T Wi þ AT4 P þ PA4 2P 5 0, Wi Ti þ 2 Wi 5 0, where
O A þ ci FC Ti ¼ , ci FC AþBK
A A4 ¼ O
and i, i ¼ 2, 3, . . . , N, are the eigenvalues of Laplacian matrix L. (2) Take min ¼ maxi¼2,3, . . . , N{ i}. Then, min is the minimum value of .
O , A non-zero
Remark 7: In Xia and Cao (2009), Cai, Liu, Xu, and Sun (2009) and Wang, Hao, and Zuo (2010), some interesting state-based intermittent feedback methods are proposed and used to analyse the synchronisation behaviours of coupled complex dynamical networks, which is a closely related topic with consensus for multi-agent systems. However, from the perspective of control theory, the states of a dynamical system are internal information which is difficult or impossible to simultaneously obtain. In contrast, protocol (3) is designed based only on the relative outputs of neighboring agents, which is more practical.
3.2 Extensions In the last subsection, consensus in multi-agent systems with a connected communication topology is studied. Notice that the communication topology may be disconnected in real multi-agent systems due to external disturbances and/or sensing range limitations. It is thus interesting and important to further investigate consensus in multi-agent systems with an arbitrarily given communication topology which may not be connected. Motivated by the works reported in Li, Wang, and Chen (2004), Xiang, Liu, Chen, Chen, and Yuan (2007) and Yu, Chen, and Lu¨ (2009a) and based on the analysis given in the last subsection, a pinningbased distributed control method is utilised here to guarantee consensus in the multi-agent system (2) under an arbitrarily given communication topology with discontinuous observations. Suppose that the communication topology of the multi-agent system (2) is given by G0 , with node set {1, 2, . . . , N}. Lemma 1 implies that changing the order of the node indexes of G0 will yield a new graph G with the Laplacian matrix L in the form of (1). To realise this transformation, an algorithm is given below. Algorithm 2: (1) Set NS ¼ {1, 2, . . . , N} and m ¼ 0. (2) Arbitrarily select a node i from NS, and use the depth-first search algorithm Tarjan (1972) to find the component CC(i) of graph G0 containing node i. Let m ¼ m þ 1 and NS ¼ NS\ NSCCðiÞ, where NSCC(i) denotes the node set of CC(i), NSCCðiÞ [ NSCCðiÞ ¼ f1, 2, . . . , Ng and NSCCðiÞ \ NSCCðiÞ ¼ ;. (3) Check the condition NS ¼ ;; if not, re-perform step (2); if so, go to step (4). (4) Arrange the m components of G0 in a sizedescending order. Then, relabel the nodes from the first component to the last one so as to obtain the graph G.
International Journal of Control Note that G0 and G are isomorphic to each other. And, under a given protocol (3), consensus in the closed-loop multi-agent system (2) with topology G0 can be achieved if and only if consensus in the closedloop multi-agent system (2) with topology G can be achieved. It is easy to check that the Laplacian matrix L of G is in the form of (1).
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In the following, consensus in multi-agent system (2) with topology G is discussed. Obviously, consensus cannot be achieved if G is disconnected. To guarantee consensus in system (2) with an arbitrarily given topology, a virtual leader labeled N þ 1 is introduced whose dynamics are given as follows: x_ Nþ1 ðtÞ ¼ AxNþ1 ðtÞ þ BuNþ1 ðtÞ, yNþ1 ðtÞ ¼ CxNþ1 ðtÞ,
ð23Þ
where xNþ1(t) 2 Rn is the state, uNþ1(t) 2 Rm is the control input, and yNþ1(t) 2 Rp is the measured output. Here, the virtual leader plays the role of a command generator providing a reference state for the followers to track. Thus, it is assumed that uNþ1(t) 0 and vNþ1(t) 0, i.e. the states of the virtual leader evolves without being effected by the followers, and the virtual leader is no need to observe the stats or outputs of any followers. It is furthermore assumed that only a subset of agents, called pinned agents, have access to the outputs of the virtual leader, but intermittently. For notational convenience, let P be the set of pinned agents and G be the augmented graph with adjacency matrix A ¼ ½aij ðNþ1ÞðNþ1Þ , where a(Nþ1)(Nþ1) ¼ 0, ai(Nþ1) 0, and ai(Nþ1) 4 0 if and only if i 2 P. The objective here is to find an appropriate control protocol for system (2) to achieve consensus in the sense of ki(t) Nþ1(t)k ¼ 0, 8i ¼ 1, 2, . . . , N, where Nþ1(t) ¼ (xNþ1(t)T, vNþ1(t)T)T. To do so, a pinningbased distributed protocol based on (3) is presented for each follower i as follows: v_i ðtÞ ¼ Avi ðtÞ þ Bui ðtÞ þ cF
N þ1 X
aij Cðvi ðtÞ vj ðtÞÞ
ðyi ðtÞ yj ðtÞÞ , ui ðtÞ ¼ Kvi ðtÞ, t 2 ½k!, k! þ Þ, v_i ðtÞ ¼ Avi ðtÞ, t 2 ½k! þ , ðk þ 1Þ!Þ,
Lemma 8: Suppose that for each i 2 {1, 2, . . . , m}, there exists at least one node ji such that ji 2 {V(i) \ P}. Then, matrix b L 4 0, where b L ¼ L þ , L is the Laplacian matrix of G, and ¼ diag{a1(Nþ1), a2(Nþ1), . . . , aN(Nþ1)} 2 RNN. Proof: Since, for each i 2 {1, 2, . . . , m}, there exists at least one node ji such that ji 2 {V(i) \ P}, it thus follows from Corollaries 1 and 2 in Ren, Beard, and McLain (2005) that L has a simple zero eigenvalue and all the other eigenvalues have positive real parts, where L is the Laplacian matrix of the augmented graph G. Taking M ¼ (a1(Nþ1), a2(Nþ1), . . . , aN(Nþ1))T 2 RN, one has " # b L M L¼ : ð25Þ 0T 0 where b L ¼ L þ . Thus, all the eigenvalues of b L have positive real parts. Since b L is symmetric, b L is positivedefinite. Based on the above analysis, an algorithm is given here to construct protocol (24). Algorithm 3: Suppose that the communication topology G0 has m component and that (A, B, C) is both stabilisable and detectable. Then, the consensus protocol (24) can be designed as follows. (1) Using Algorithm 2 to relabel the node indexes of G0 so as to get a graph G whose Laplacian matrix L has the form of (1). Then, pin m different nodes j1, j2, . . . , jm, where ji 2 V(i), and V(i) is the node set of the ith component of G, i ¼ 1, 2, . . . , m. (2) Choose & 0 4 0, and select a feedback gain matrix K by Ackermann’s formula, such that the real parts of the poles of A þ BK are smaller than & 0. (3) Solve the LMI AT Q þ QA 2CT C þ 2&0 Q 5 0
ð26Þ
to obtain one solution Q 4 0. Then, take the feedback gain matrix F ¼ Q1CT. (4) Take the coupling strength c 1/0, where 0 is the smallest eigenvalue of matrix b L.
j¼1
ui ðtÞ ¼ 0,
101
To derive another main result, the following lemmas are introduced for which the proofs are simple therefore omitted. k 2 N,
ð24Þ
where vi(t) 2 Rn is the state of the observer embedded in agent i, c 4 0 is the coupling strength, and F 2 Rnp and K 2 Rmn are the feedback matrices. Before moving forward, the following lemma is introduced. Let V(i) be the node set of the ith (1 i m) connected component of G.
Lemma 9: The maximum real parts of the eigenvalues of matrices Ti are less than & 0, where A þ ci FC O Ti ¼ , ci FC A þ BK F ¼ Q1CT, Q is a solution of LMI (26), and i, i ¼ 1, 2, . . . , N, are the eigenvalues of b L.
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Lemma 10:
Ti T Wi þ Wi Ti þ 2&Wi 5 0,
ð27Þ
where & 5 & 0, Ti ¼
A þ ci FC
O
ci FC
A þ BK
,
L. and i, i ¼ 1, 2, . . . , N, are the eigenvalues of b
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Now, it is to present another main result of this article. Theorem 2: For the multi-agent system (2) with a communication topology G0 , the protocol (24) constructed by Algorithm 3 solves the consensus problem if the communication rate /! 4 /(& þ ) þ ln%/ [(& þ )!], where 0 5 & 5 & 0, % ¼ maxi¼1, 2, . . . , N {max(Wi)/min(P), max(P)/min(Wi)}, 4 0, 0 is given in Lemma 7, and matrices P and Wi are positivedefinite solutions of LMIs (18) and (27), respectively. Proof: Take x i ðtÞ ¼ xi ðtÞ xNþ1 ðtÞ and vi ðtÞ ¼ vi ðtÞ vNþ1 ðtÞ, where i ¼ 1, 2, . . . , N. Obviously, consensus in (2) can be achieved if and only if kx i ðtÞk ! 0 and kvi ðtÞk ! 0, for all i ¼ 1, 2, . . . , N. Based on the above analysis and according to (24), one gets v_i ðtÞ ¼ Avi ðtÞ þ Bui ðtÞ þ cF
N X
b lij C vj ðtÞ x j ðtÞ ,
j¼1
ui ðtÞ ¼ Kvi ðtÞ, v_i ðtÞ ¼ Avi ðtÞ, ui ðtÞ ¼ 0,
t 2 ½k!, k! þ Þ,
t 2 ½k! þ , ðk þ 1Þ!Þ,
k 2 N,
ð28Þ h i where b L¼ b lij . NN T T Let i ðtÞ ¼ x i ðtÞ , vi ðtÞT , i ¼ 1, 2, . . . , N. Then, it follows from (2) and (3) that N P _i ðtÞ ¼ A1 i ðtÞ þ c b lij Hj ðtÞ,
t 2 ½k!, k! þ Þ,
j¼1
_i ðtÞ ¼ A2 i ðtÞ,
t 2 ½k! þ , ðk þ 1Þ!Þ,
k 2 N, ð29Þ
where
A BK A1 ¼ , O AþBK O O H¼ : FC FC
A A2 ¼ O
O , A
Similarly to the proof of Theorem 1, the rest of the proof can be completed. Remark 8: The minimum admissible communication rate for achieving consensus under a given
communication topology with fixed parameters & and can be obtained by minimising the parameter % in Theorem 2. And, the minimum % can be obtained by solving the following optimisation problem: (1) Minimise %i subject to: Wi 4 0, P 4 Wi, P 5 %iWi, AT4 P þ PA4 Ti T Wi þ Wi Ti þ 2&Wi 5 0, 2P 5 0, where A þ ci FC A O O , Ti ¼ , A4 ¼ O A ci FC A þ BK and i, i ¼ 1, 2, . . . , N, are the eigenvalues of b L. (2) Take %min ¼ maxi¼1, 2, . . . , N{%i}. Then, %min is the minimum value of %.
4. Simulation examples In this section, two simulation examples are provided to verify the theoretical analysis. Example 1: Take each agent in a multi-agent system to be a two-mass-spring system with a single force input (Zhang, Lewis, and Qu 2012), whose dynamics are described by (2) with 0 1 0 1 0 1 0 0 xi1 ðtÞ B x ðtÞ C B k1 k2 0 k2 0 C B i2 C B m C m1 xi ðtÞ ¼ B C, A ¼ B 1 C, 0 0 1A @ xi3 ðtÞ A @ 0 k2 2 0 k 0 xi4 ðtÞ m2 m2 0 1 0 B 1 C B C B ¼ B m1 C, ð30Þ @0A 0 where m1 and m2 are two masses, and k1 and k2 are spring constants. Furthermore, take the output matrix 0 1 1 0 0 0 B C C ¼ @ 0 1 0 0 A, 0 0 m1 ¼ 1.2 kg, k2 ¼ 1.0 N/m.
m2 ¼ 1.0 kg,
1
0 k1 ¼ 1.4 N/m,
and
Some simple calculations show that (A, B, C) is controllable and observable. Consider a group of four agents with undirected communication topology G1 as shown in Figure 1, where the weights are indicated on the edges. Obviously, G1 is connected. An observer-type of consensus protocol in the form of (3) is designed according to Algorithm 1. Take ! ¼ 4 and 0 ¼ 2.
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International Journal of Control 0.8
1
1.5
2
x11 (t) x21 (t) x31 (t) x41 (t)
1.0
0.4
4
3
xi1(t), i=1,2,3,4.
1 0.5 0 −0.5
Figure 1. Communication topology G1, in Example 1.
−1 −1.5 0
With communication
5
10 t
15
20
Without communication 3.2
3.2
4 x 12 (t) x 22 (t) x 32 (t) x 42 (t)
3 4
Figure 2. Intermittent communication.
Then, some calculations give the feedback gain matrices in (3) as K ¼ (27.12, 9.72, 3.32, 35.04) and 0 1 3:2838 0:6105 0:0594 B 0:6105 3:9534 0:6675 C B C ð31Þ F ¼B C: @ 0:0594 0:6675 8:1926 A 0:1739
0:0167
2
time
8
20:7378
According to Lemmas 6 and 7, one may take ¼ 1.15, ¼ 1.5. Letting c ¼ 2.75 and solving the optimisation problem in Remark 6 gives min ¼ 11. Thus, the minimum communication rate is 79.22%. Take ¼ 3.20, which means that the communication rate is 80%, see Figure 2 for illustration. According to Theorem 1, the states of all agents in system (2) will converge to the same value. The state trajectories of the agents are shown in Figures 3–6, respectively. Use vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3 uX kxi ðtÞ x4 ðtÞk2 E1 ðtÞ ¼ t i¼1
xi2(t), i=1,2,3,4.
0
1 0 −1 −2 −3 −4 −5 −6 0
5
10 t
15
20
Figure 4. Consensus of trajectories of xi2(t), i ¼ 1, 2, 3, 4, in Example 1. 1.2 x13(t) x23(t) x33(t) x43(t)
1 0.8 xi3(t), i=1,2,3,4.
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Figure 3. Consensus of trajectories of xi1(t), i ¼ 1, 2, 3, 4, in Example 1.
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
0
5
10 t
15
20
to denote the consensus errors of system (2) under protocol (3). Figure 7 demonstrates that consensus is indeed achieved.
Figure 5. Consensus of trajectories of xi3(t), i ¼ 1, 2, 3, 4, in Example 1.
Example 2: Consider a group of seven agents with a communication topology G2 as shown in Figure 8, where the agents 1, 2, 3 belong to the first component and agents 4, 5, 6, 7 belong to the second component. The agent dynamics and the parameters are the same
as those in Example 1. To achieve consensus, a virtual leader labeled by 8 is introduced. Then, an observertype of consensus protocol in the form of (24) is designed according to Algorithm 3. According to
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2 x 14(t) x 24(t) x 34(t) x 44(t)
xi1(t), i=1,2,...,8.
xi4(t), i=1,2,3,4.
1 0.5 0
−0.5 −1
1 0.5 0 −0.5 −1 −1.5 −2
0
5
10 t
15
20
Figure 6. Consensus of trajectories of xi4(t), i ¼ 1, 2, 3, 4, in Example 1.
0
5
10 t
15
20
Figure 9. Consensus of trajectories of xi1(t), i ¼ 1, 2, . . . , 8, in Example 2.
5 9
3
Consensus errors E1(t) xi2(t), i=1,2,...,8.
7 6 5 4
1 0 −1 −2 −3
2
−4
1
−5 0
5
10 t
Figure 7. Trajectories Example 1.
of
15
consensus
8
4
1
5
errors
E1(t),
in
1 1
1.2
1.75
2 4
3
0
5
10 t
15
20
20
5
7
6
2
3
0
x12 (t) x22 (t) x32 (t) x42 (t) x52 (t) x62 (t) x72 (t) x82 (t)
4
8
E1(t)
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x 11(t) x 21(t) x 31(t) x 41(t) x 51(t) x 61(t) x 71(t) x 81(t)
1.5
1.5
2
Figure 8. Communication topology G2, in Example 2.
step (1) of Algorithm 3, agents 1 and 7 are chosen as the pinned agents, i.e. P ¼ {1, 7}. Simple calculations yield that 0 ¼ 0.5493, where 0 is the smallest eigenvalue of b L given in (25). Take & ¼ 1.15, ¼ 1.5 and ! ¼ 4. Selecting c ¼ 2.75 and solving the optimization problem in Remark 8 gives %min ¼ 10.70. According to
Figure 10. Consensus of trajectories of xi2(t), i ¼ 1, 2, . . . , 8, in Example 2.
Theorem 2, the minimum communication rate is 78.96%. Take ¼ 3.20, which means that the communication rate is 80%, see Figure 2 for illustration. The feedback matrices F and K are taken to be the same as those in Example 1. Then, according to Theorem 2, the states of the agents in system (2) will approach those of the virtual leader. The state trajectories of agents are shown in Figures 9–12, respectively. Use vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 7 uX kxi ðtÞ x8 ðtÞk2 E2 ðtÞ ¼ t i¼1
to denote the consensus errors of system (2) under protocol (24). Figure 13 demonstrates that consensus is indeed achieved.
5. Conclusions In this article, the consensus problem for multi-agent systems with general linear node dynamics and
International Journal of Control 1.5
x13 (t) x23 (t) x33 (t) x43 (t) x53 (t) x63 (t) x73 (t) x83 (t)
xi3(t), i=1,2,...,8.
1 0.5 0 −0.5 −1 −1.5
0
5
10 t
15
20
Figure 11. Consensus of trajectories of xi3(t), i ¼ 1, 2, . . . , 8, in Example 2. x 14(t) x 24(t) x 34(t) x 44(t) x 54(t) x 64(t) x 74(t) x 84(t)
xi4(t), i=1,2,...,8.
1.5 1 0.5 0 −0.5
been proved that consensus in the closed-loop multiagent systems can be guaranteed under a fixed connected undirected topology if the communication rate is larger than a threshold value. Furthermore, consensus for multi-agent systems with a disconnected communication topology has been studied from a pinning-based distributed control approach. The effectiveness of the theoretical analysis has been verified by numerical simulations. Future work will focus on consensus for multi-agent systems with higher-order nonlinear dynamics and discontinuous output measurements.
Acknowledgements The authors sincerely thank the Associate Editor and all the anonymous reviewers for their valuable comments which helped them improve the manuscript of this paper. This work was supported by the Hong Kong Research Grants Council under the GRF Grant CityU 1114/11E, the National Nature Science Foundation of China under Grants 60974078, 90916003, 61104153, the China Postdoctoral Science Foundation under Grants 20100480211, 201104059 and the Information Processing and Automation Technology Prior Discipline of Zhejiang Province-Open Research Foundation under Grant 20120801.
−1 −1.5
0
5
10 t
15
References
20
Figure 12. Consensus of trajectories of xi4(t), i ¼ 1, 2, . . . , 8, in Example 2. 9 Consensus errors E2(t)
8 7 6 E2(t)
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2
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5 4 3 2 1 0
0
5
10 t
15
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Figure 13. Trajectories of consensus errors E2(t), in Example 2.
discontinuous output measurements has been studied. To achieve consensus, a class of observer-type of protocols have been proposed. By using tools from switching systems theory and matrix analysis, it has
Boyd, S., Ghaoui, L.E., Ferion, E., and Balakrishnan, V. (1994), Linear Matrix Inequalities in System and Control Theory, Philadelphia, PA: SIAM. Cai, S., Liu, Z., Xu, F., and Sun, J. (2009), ‘Periodically Intermittent Controlling Complex Dynamical Networks with Time-varying Delays to a Desired Orbit’, Physics Letters A, 373, 3846–3854. Gazi, V., and Passino, K.M. (2003), ‘Stability Analysis of Swarms’, IEEE Transactions on Automatic Control, 48, 692–697. Godsil, C., and Royle, G. (2001), Algebraic Graph Theory, New York: Springer-Verlag. Hong, Y., Chen, G., and Bushnell, L. (2008), ‘Distributed Observers Design for Leader-following Control of Multiagent Networks’, Automatica, 44, 846–850. Huang, L. (1984), Linear Algebra in System and Control Theory, Beijing, China: Science Press. Jadbabaie, A., Lin, J., and Morse, A.S. (2003), ‘Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbour Rules’, IEEE Transactions on Automatic Control, 48, 988–1001. Li, Z., Duan, Z., and Chen, G. (2011), ‘Dynamic Consensus of Linear Multi-agent Systems’, IET Control Theory and Applications, 5, 19–28. Li, Z., Duan, Z., Chen, G., and Huang, L. (2010), ‘Consensus of Multiagent Systems and Synchronization of Complex Networks: A Unified Viewpoint’, IEEE Transactions on Circuits Systems I: Regular Papers, 57, 213–224.
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Distributed consensus control for linear multi-agent systems with discontinuous observations Guanghui Wena, Zhongkui Lib, Zhisheng Duana* and Guanrong Chenac a
Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China; b School of Automation, Beijing Institute of Technology, Beijing 100081, China; cDepartment of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China
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(Received 25 May 2012; final version received 5 August 2012) This article addresses the distributed consensus problem of linear multi-agent systems with discontinuous observations over a time-invariant undirected communication topology. Under the assumption that each agent can only intermittently share its outputs with the neighbours, a class of distributed observer-type of protocols are designed and utilised to achieve consensus. By using appropriate matrix decomposition, it is shown that consensus in the closed-loop multi-agent systems under a connected topology can be converted to the simultaneous asymptotic stability of a set of switching systems whose dimensions are the same as each agent. From a multiple Lyapunov functions approach, it is proved that there exists a protocol to guarantee consensus if the communication time rate is larger than a threshold value. Furthermore, a distributed pinning control method is employed to solve the consensus problem on an arbitrary given topology which needs not be connected. Particularly, the questions of what kind of agents and at least how many agents should be pinned are addressed. The effectiveness of the analytical results is finally verified by numerical simulations. Keywords: multi-agent system; consensus; pinning control; discontinuous observation; lyapunov function
1. Introduction Recently, distributed cooperative control of multiagent systems has received considerable attention from various scientific communities due to the growing interest in understanding the intriguing animal group behaviours, such as swarming (Gazi and Passino 2003) and flocking (Su, Wang, and Lin 2009; Wen, Duan, Li, and Chen 2012b), and their potential applications in formation control of satellites (Smith and Hadaegh 2005), teaming of multi-robotics (Ren and Sorensen 2008), and design of sensor networks (Yu, Chen, Wang, and Yang 2009b). Among the numerous research topics in distributed control of multi-agent systems, consensus problem is of particular importance, which refers to designing an appropriate protocol based only on the local relative information among neighbouring agents to guide all agents to reach an agreement (see survey papers Olfati-Saber, Fax, and Murray (2007), Ren, Beard, and Atkins (2007a) and the references therein). Vicsek, Cziro´k, Ben-Jacob, Cohen, and Shochet (1995) introduced a simple yet effective discrete-time model for phase transition of a group of autonomous agents and numerically investigated the angle
*Corresponding author. Email: [email protected] ISSN 0020–7179 print/ISSN 1366–5820 online ß 2013 Taylor & Francis http://dx.doi.org/10.1080/00207179.2012.719637 http://www.tandfonline.com
consensus behaviour of the multi-agent system. By using the tools from the algebraic graph theory, a theoretical explanation of the consensus behavior observed in Vicsek et al. (1995) was first given in Jadbabaie, Lin, and Morse (2003). A general framework of the consensus problem for a network of agents with single-integrator dynamics under a fixed or a switching topology and possible communication time delays was suggested and studied in Olfati-Saber and Murray (2004). The consensus conditions derived in Olfati-Saber and Murray (2004) were further relaxed in Ren and Beard (2005) by proving that consensus in multi-agent systems with single-integrator dynamics can be achieved if and only if the time-varying network topology contains a directed spanning tree frequently enough as the network evolves over time. Meanwhile, the consensus problem for multi-agent systems with second- and higher-order dynamics were addressed (Ren and Atkins 2007; Xie and Wang 2007; Hong, Chen, and Bushnell 2008; Wen, Duan, Yu, and Chen 2012d; Ren, Moore, and Chen 2007b; Zhang and Lewis 2012). Note that most of the above-mentioned works are concerned with the case where the agents are governed by integrator-type dynamics.
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However, multi-agent systems with general linear node dynamics are more popular (Ma and Zhang 2010; Li, Duan, Chen, and Huang 2010; Zhang, Lewis, and Das 2011; Li, Duan, and Chen 2011), which include networks of agents with integrator-type of dynamics as special cases. In Ma and Zhang (2010), from a static output approach, some necessary and sufficient conditions were derived for consensus of multi-agent systems with general linear node dynamics under a fixed directed topology. Consensus in multi-agent systems with general linear node dynamics under a fixed directed topology was investigated with observertype protocols appropriately designed in Li et al. (2010), Zhang et al. (2011), Li et al. (2011). Most of the above-mentioned results on the consensus problem in multi-agent systems with general linear dynamics are obtained based on the assumption that information is transmitted continuously among the agents, i.e. each agent has to share its state or output information with its neighbours all the time. However, this may not always be the case in reality. Sometimes, mobile agents can only communicate with their neighbours over some disconnected time intervals due to, for instance, temporary sonar equipment failures or the presence of communication obstacles, even if the distances among them are less than the communication radius. Yet, equipment failures may be recovered through repairing and communication obstacles may be bypassed as the system evolves in time. Motivated by these facts and based on the works reported in Wen, Duan, and Chen (2012a), Wen, Duan, Li, and Chen (2012c), the consensus problem for multi-agent systems with general linear node dynamics based on intermittent observations is studied in this article, where the ideal assumption that agents could transmit their output information to their neighbours at all times is removed. For convenience of analysis, the communication topology among the agents is assumed to be undirected. For achieving consensus, a new class of consensus protocols are proposed and analysed. By using tools from switching systems theory, it is theoretically shown that consensus in a closed-loop multi-agent system with a connected topology can be ensured if the communication rate is larger than a threshold value. The analytical expression of the threshold value is also explicitly given. By using a distributed pinning-based control method, the results are then extended to consensus in multi-agent systems with an arbitrary topology which needs not be connected. In particular, the questions of what kind of agents and at least how many agents should be pinned for achieving consensus are addressed and answered. Numerical examples are finally given to verify the theoretical analysis.
The rest of this article is organised as follows. Some preliminaries and the model formulation are presented in Section 2. Consensus problem for multi-agent systems with discontinuous observations under a time-invariant connected topology is studied in Section 3. Some extensions are given in Section 4. In Section 5, several numerical simulations are provided for illustration. Conclusions are finally drawn in Section 6. Throughout this article, let N and R be the set of natural and real numbers, respectively, and Rnn be the sets of n n real matrices. Let In (On) be the n n identity (zero) matrices, and 1n (0n) be the n-dimensional column vector with all entries equal to one (zero). Matrices, if not explicitly stated, are assumed to have compatible dimensions. The matrix inequality A 4 B means that both A and B are square symmetric matrices and that A B is positive-definite. diag{a1, a2, . . . , aN} represents a diagonal matrix with ai, i ¼ 1, 2, . . . , N, being its diagonal elements. Notations and k k represent the Kronecker product and the Euclidian norm, respectively.
2. Preliminaries and formulation of the model In this section, some preliminaries and the model formulation for consensus in multi-agent systems with discontinuous observations are introduced.
2.1 Preliminaries An undirected graph G is a pair of (V, E), where V ¼ {1, 2, . . . , N} is a node set and E {(i, j ), i, j 2 V} is an edge set in which an edge is represented by an unordered pair of distinct nodes. Two nodes i, j are adjacent or neighboring, if (i, j) is an edge of graph G, i.e. (i, j) 2 E. A path on G from node i1 to node is is a sequence of ordered edges of the form (ik, ikþ1), k ¼ 1, 2, . . . , s 1. An undirected graph is connected if there exists a path between every pair of distinct nodes, otherwise is disconnected. Only simple graphs are considered here, i.e. multiple edges and self-loops are forbidden in G. A connected subgraph of G, which is maximal, is called a component of G. The adjacency matrix A ¼ [aij]NN of a graph G is defined by aii ¼ 0 for i ¼ 1, 2, . . . , N, and aij ¼ aji 4 0 for (i, j) 2 E but 0 otherwise. The Laplacian matrix is defined as lij ¼ aij, i 6¼ j, and L ¼ [lP ij]NN a lii ¼ N k¼1 ik for i ¼ 1, 2, . . . , N. For an undirected graph G, both its adjacency matrix and Laplacian matrix are symmetric. The following lemmas can be found in graph theory textbooks (e.g. Godsil and Royle 2001).
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International Journal of Control Lemma 1: Suppose that an undirected graph G has m components. Then, there exist a permutation matrix W of order N, such that 2 3 e O L11 O 6 7 L22 O 7 6 O e 7, ð1Þ WT LW ¼ 6 .. .. 6 .. 7 4 . . O 5 . O O e Lmm where e L11 2 Rq1 q1 , e L22 2 Rq2 q2 , . . . , e Lmm 2 Rqm qm are Pm symmetric matrices with zero row sums, with j¼1 qj ¼ N and 1 q1 qm N. Furthermore, R(L) ¼ N m, where R(L) represents the rank of L.
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Remark 1: Any undirected graph G contains at least one component and at most N components. Thus, according to Lemma 1, one has 0 R(L) N 1. Lemma 2: Consider a connected undirected graph G. Then, 0 is a simple eigenvalue of its Laplacian matrix L and all the other eigenvalues of L are positive real numbers. 2.2 Formulation of the model Consider a network of identical agents with linear or linearised dynamics, described by x_ i ðtÞ ¼ Axi ðtÞ þ Bui ðtÞ, ð2Þ yi ðtÞ ¼ Cxi ðtÞ, i ¼ 1, 2, . . . , N, where xi(t) 2 Rn is the state, ui(t) 2 Rm is the control input, yi(t) 2 Rp is the measured output, and A, B and C are constant real matrices with compatible dimensions. The communication topology among the N agents is represented by an undirected graph G consisting of the node set V ¼ {1, 2, . . . , N} and the edge set E {(i, j), i, j 2 V}. An existing edge (i, j) (i 6¼ j) means that agents i and j can obtain information from each other. In some real situations, agents may only sense the outputs of their neighbours over some disconnected time intervals due to the unreliability of communication channels, failure of physical devices, etc. Motivated by this observation, the following distributed observer-type of consensus protocol with discontinuous dynamic output measurements is proposed for each agent i: v_i ðtÞ ¼ Avi ðtÞ þ Bui ðtÞ þ cF ðyi ðtÞ yj ðtÞÞ , ui ðtÞ ¼ Kvi ðtÞ, v_i ðtÞ ¼ Avi ðtÞ, ui ðtÞ ¼ 0,
N X
aij Cðvi ðtÞ vj ðtÞÞ
where vi(t) 2 Rn is the state of the observer embedded in agent i, c 4 0 is the coupling strength, F 2 Rnp and K 2 Rmn are feedback matrices, A ¼ [aij]NN is adjacency matrix of graph G, and ! 4 4 0. Let i(t) ¼ (xi(t)T, vi(t)T)T. Then, it follows from (2) and (3) that _i ðtÞ ¼ A1 i ðtÞ þ c
N P
lij Hj ðtÞ,
t 2 ½k!, k! þ Þ,
j¼1
_i ðtÞ ¼ A2 i ðtÞ,
t 2 ½k! þ , ðk þ 1Þ!Þ,
ð4Þ
k 2 N,
where
, O AþBK O O H¼ , FC FC
A1 ¼
A
BK
A2 ¼
A
O
O
A
,
and L ¼ [lij]NN is the Laplacian matrix of graph G. Remark 2: The consensus problem for multi-agent systems with general linear node dynamics based on intermittent relative state information has been studied recently in Wen et al. (2012a), using the state information of agents which are hard or impossible to obtain in practice. In contrast, the present protocol (3) depends only on the relative output information of neighboring agents. Definition 1: The consensus problem of multi-agent system (2) is solved by protocol (3) if, for any initial conditions, the states of system (4) satisfy lim i ðtÞ j ðtÞ ¼ 0,
t!1
8 i, j ¼ 1, 2, . . . , N:
ð5Þ
Remark 3: From Definition 1, consensus in multiagent system (2) is solved by protocol (3) if and only if both the states of agents and the protocols embedded in agents asymptotically approach the same values, respectively. Finally, the following lemmas (referring to Boyd, Ghaoui, Ferion, and Balakrishnan 1994) are introduced. Lemma 3: The maximum real parts of eigenvalues of a matrix A 2 Rnn is less than , where 4 0, if and only if there exists a matrix P ¼ PT 4 0 such that AT P þ PA þ 2P 5 0:
j¼1
Lemma 4: For matrices A, B, C, and D of appropriate dimensions, one has
t 2 ½k!, k! þ Þ,
t 2 ½k! þ , ðk þ 1Þ!Þ,
k 2 N,
ð3Þ
(1) (A) B ¼ A (B) ¼ (A B), 8 2 R; (2) (A þ B) C ¼ A C þ B C; (3) (A B)(C D) ¼ (AC) (BD).
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3. Main results In this section, the mains results of this article are described and proved.
3.1 Consensus in multi-agent systems under a connected topology In this subsection, distributed consensus in multi-agent system (2) under an undirected connected topology is addressed.
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Assumption 1: The undirected communication topology G is connected. Under Assumption 1, it follows from Lemma 2 that ¼ (1/N )1N is the left eigenvector of the Laplacian matrix L associated with the zero eigenvalue. Let s(t) (s1(t)T, s2(t)T, . . . , sN(t)T)T, where si ðtÞ ¼ i ðtÞ P¼ N 1 j¼1 j ðtÞ, i ¼ 1, 2, . . . , N. Then, N sðtÞ ¼ IN 1N T In ðtÞ, ð6Þ where (t) ¼ (1(t)T, 2(t)T, . . . , N(t)T)T. It is easy to verify that s(t) ¼ 0 if and only if 1(t) ¼ 2(t) ¼ ¼ N(t), for all t 0. It then follows from (3) and (6) that s_ðtÞ ¼ ðIN A1 þ cL HÞsðtÞ, t 2 ½k!, k! þ Þ, s_ðtÞ ¼ ðIN A2 ÞsðtÞ, t 2 ½k! þ , ðk þ 1Þ!Þ, ð7Þ where
A BK , A1 ¼ O AþBK
O O H¼ , FC FC
A2 ¼
A O
O , A
and L ¼ [lij]NN is the Laplacian matrix of graph G. Let Y1 2 RN(N1), Y2 2 R(N1)N, T 2 RNN and a diagonal matrix D 2 R(N1)(N1) be such that T
1 T ¼ ð1N , Y1 Þ, T ¼ , Y2 ! 0 0TN1 T1 LT ¼ ¼ , ð8Þ 0N1 D where the diagonal entries of D are the non-zero eigenvalues of Laplacian matrix L. By using the linear transformation ð9Þ ðtÞ ¼ T1 I2n sðtÞ with (t) ¼ (1(t)T, 2(t)T, . . . , N(t)T)T, it follows from Lemma 4 and (7) that _ ¼ ðIN A1 þ c HÞðtÞ, t 2 ½k!, k! þ Þ, ðtÞ _ ¼ ðIN A2 ÞðtÞ, t 2 ½k! þ , ðk þ 1Þ!Þ, k 2 N, ðtÞ ð10Þ
where A1, A2, H are defined in (7) and is given in (8). Note that 1(t) 0, for all t 0. Thus, the consensus problem for multi-agent system (2) is solved by protocol (3) if and only if i(t), i ¼ 2, 3, . . . , N, converge asymptotically to zero, which is in turn equivalent to that the following N 1 systems z_i ðtÞ ¼ ðIN A1 þ ci HÞzi ðtÞ, t 2 ½k!, k! þ Þ, z_i ðtÞ ¼ ðIN A2 Þzi ðtÞ, t 2 ½k! þ , ðk þ 1Þ!Þ, k 2 N, ð11Þ are simultaneously asymptotically stable, where i, i ¼ 2, 3, . . . , N, are the non-zero eigenvalues of the Laplacian matrix L. Taking linear transformation
I I zi ðtÞ, i ¼ 2, 3, . . . , N, ð12Þ i ðtÞ ¼ O I and using (11), one obtains _ i ðtÞ ¼ Ti i ðtÞ, _ i ðtÞ ¼ A2 i ðtÞ,
t 2 ½k!, k! þ Þ, t 2 ½k! þ , ðk þ 1Þ!Þ,
k 2 N, ð13Þ
where
A þ ci FC O Ti ¼ , ci FC A þ BK
A A2 ¼ O
O , A
and i, i ¼ 2, 3, . . . , N, are the non-zero eigenvalues of the Laplacian matrix L. From the above analysis, one has converted the consensus problem for multi-agent system (2) with protocol (3) to the simultaneously asymptotical stability problem of N 1 switching systems given by (13), which have the same low dimension as a single agent. One can see that the effects of the communication topology on the consensus are characterised by the non-zero eigenvalues of the Laplacian matrix L. Next, an algorithm is presented to construct protocol (3). Algorithm 1: Under Assumption 1 and the condition that (A, B, C) is both stabilisable and detectable, the consensus protocol (3) can be designed as follows. (1) Choose 0 4 0, and select feedback gain matrix K by Ackermann’s formula, such that the real parts of the poles of A þ BK are smaller than 0. (2) Solve the following linear matrix inequality (LMI): AT Q þ QA 2CT C þ 2 0 Q 5 0,
ð14Þ
to obtain one solution Q 4 0. Then, let the feedback gain matrix be F ¼ Q1CT.
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International Journal of Control (3) Take the coupling strength c 1/2, where 2 is the second smallest eigenvalue of the Laplacian matrix L. Lemma 5: The maximum real parts of eigenvalues of matrices Ti are less than 0, where Ti ¼ 1
A þ ci FC
O
ci FC
A þ BK
,
T
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F ¼ Q C , Q is a solution of LMI (14), and i, i ¼ 2, 3, . . . , N, are the non-zero eigenvalues of L. Proof: Similarly to the proof of Proposition 1 in Li et al. (2011), this lemma can be proved. For the readers’ convenience, a sketched proof is given as follows. Let Q 4 0 be a solution of LMI (14). Take F ¼ Q1CT. Then, one gets ðA þ ci FCÞQ1 þ Q1 ðA þ ci FCÞT þ 2 0 Q1 ¼ AQ1 þ Q1 AT 2ci Q1 CT CQ1 þ 2 0 Q1
Pn Proof: With 0 ¼ maxj¼1,2,..., n i¼1 aij , the lemma can be proved by using the Gersˇ gorin disc theorem (Huang 1984). The following theorem is one main result of this articler. Theorem 1: For the multi-agent system (2), under Assumption 1 and moreover (A, B, C) is both stabilisable and detectable, the protocol (3) constructed by Algorithm 1 solves the consensus problem if the communication time rate /! 4 /( þ ) þ ln /[( þ )!], where 0 5 5 0, ¼ maxi¼2,3, . . . , N {max(Wi)/min(P), max(P)/min(Wi)}, 4 0, and matrices Wi and P are the positive-definite solutions of LMIs (17) and (18), respectively. Proof: Construct the following multiple Lyapunov function candidate for the ith switching system of (13): ( i ðtÞT Wi i ðtÞ, t 2 ½k!, k! þ Þ, VðtÞ ¼ i ðtÞT P i ðtÞ, t 2 ½k! þ , ðk þ 1Þ!Þ,
AQ1 þ Q1 AT 2Q1 CT CQ1 þ 2 0 Q1 ,
ð19Þ ð15Þ
where the last inequality of (15) is derived by using the fact c 1/2, with 2 being the second smallest eigenvalue of the Laplacian matrix L. Pre- and post-multiplying (15) by Q and its transpose, according to (14), yields ðA þ ci FCÞQ1 þ Q1 ðA þ ci FCÞT þ 2 0 Q1 5 0: ð16Þ It thus follows from Lemma 3 and (16) that the maximum real parts of the eigenvalues of matrices Ti are less than 0, where A þ ci FC O Ti ¼ : ci FC A þ BK Lemma 6: There exist some Wi 4 0 such that Ti T Wi þ Wi Ti þ 2 Wi 5 0,
Ti ¼
ð17Þ
A þ ci FC
O
ci FC
A þ BK
,
k! þ Þ,
ð20Þ
ðk þ 1Þ!Þ:
ð21Þ
where is defined in Lemma 6. Similarly, one has _ 5 2VðtÞ, VðtÞ
t 2 ½k! þ ,
Note that the switching systems (13) switches at t ¼ k! and t ¼ k! þ , k 2 N. Based on the above analysis, one obtains
5 2 e2 þ2ð!Þ Vð0Þ
Vðk!Þ 5 Vð0Þek :
It follows directly from Lemmas 3 and 5.
Lemma 7: There exists a 0 4 0, such that for all 4 0, the following LMI
O
t 2 ½k!,
where ¼ 2 2(! ) 2ln . From the condition /! 4 /( þ ) þ ln /[( þ )!], one has 4 0. By recursion, for any positive integer k, one gets
i ¼ 2, 3, . . . , N.
where A4 ¼
_ 5 2 VðtÞ, VðtÞ
¼ e Vð0Þ,
A
For t 2 [k!, k! þ ) and an arbitrarily given k 2 N, taking the time derivative of V(t) along the trajectories of system (13) gives
Vð!Þ 5 e2ð!Þ VðÞ
where 5 0,
Proof:
where the positive-definite matrices Wi and P are the solutions of (17) and (18), respectively, k 2 N, i ¼ 2, 3, . . . , N.
AT4 P þ PA4 2P 5 0, O A , has a solution P 4 0.
ð18Þ
ð22Þ
For any t 4 0, there exists an r 2 N such that r! t 5 (r þ 1)!. Then, one has VðtÞ 5 Vðr!Þe2! 5 Vð0Þer þ2! 5 e2! Vð0Þe½ =ð!þ1Þ t ,
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G. Wen et al.
i.e. VðtÞ 5 0 e t ,
for all t 0,
where 0 ¼ e2!V(0), and ¼ /(! þ 1). This indicates that the states of agents exponentially converge to the same.
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Remark 4: In Theorem 1, it is assumed that the agents can obtain the intermittent relative outputs of its neighbours periodically. However, one may extend the results to the consensus in linear multi-agent systems with aperiodically intermittent output communications by using the present approach. Similarly to the proof of Theorem 1, some corresponding theoretical results can be derived, which are omitted here for brevity. Remark 5: To successfully construct protocol (3) according to Algorithm 1, it should be firstly shown that both steps (1) and (2) of Algorithm 1 are feasible for some 0 4 0. Note that there always exist some positive scalar 0 such that steps (1) and (2) are feasible if (A, B, C) is stabilisable and detectable. It is also worth noting that both steps (1) and (2) are feasible for any given 0 4 0 if the matrix triple (A, B, C) is controllable and observable. Remark 6: In the context of multi-agent systems with intermittent observations, one interesting and important issue is what the minimum admissible communication rate is for achieving consensus for a given topology G with fixed parameters and . However, LMIs (17) and (18) are solved independently, which may introduce conservativeness in determining the admissible communication rate to satisfy the consensus condition. From the proof of Theorem 1, one can see that the minimum admissible communication rate, under a given communication topology with fixed parameters and , can be obtained by minimising in Theorem 1. Actually, the minimum can be obtained by solving the following optimisation problem: (1) Minimise i subject to: Wi 4 0, P 4 Wi, P 5 iWi, Ti T Wi þ AT4 P þ PA4 2P 5 0, Wi Ti þ 2 Wi 5 0, where
O A þ ci FC Ti ¼ , ci FC AþBK
A A4 ¼ O
and i, i ¼ 2, 3, . . . , N, are the eigenvalues of Laplacian matrix L. (2) Take min ¼ maxi¼2,3, . . . , N{ i}. Then, min is the minimum value of .
O , A non-zero
Remark 7: In Xia and Cao (2009), Cai, Liu, Xu, and Sun (2009) and Wang, Hao, and Zuo (2010), some interesting state-based intermittent feedback methods are proposed and used to analyse the synchronisation behaviours of coupled complex dynamical networks, which is a closely related topic with consensus for multi-agent systems. However, from the perspective of control theory, the states of a dynamical system are internal information which is difficult or impossible to simultaneously obtain. In contrast, protocol (3) is designed based only on the relative outputs of neighboring agents, which is more practical.
3.2 Extensions In the last subsection, consensus in multi-agent systems with a connected communication topology is studied. Notice that the communication topology may be disconnected in real multi-agent systems due to external disturbances and/or sensing range limitations. It is thus interesting and important to further investigate consensus in multi-agent systems with an arbitrarily given communication topology which may not be connected. Motivated by the works reported in Li, Wang, and Chen (2004), Xiang, Liu, Chen, Chen, and Yuan (2007) and Yu, Chen, and Lu¨ (2009a) and based on the analysis given in the last subsection, a pinningbased distributed control method is utilised here to guarantee consensus in the multi-agent system (2) under an arbitrarily given communication topology with discontinuous observations. Suppose that the communication topology of the multi-agent system (2) is given by G0 , with node set {1, 2, . . . , N}. Lemma 1 implies that changing the order of the node indexes of G0 will yield a new graph G with the Laplacian matrix L in the form of (1). To realise this transformation, an algorithm is given below. Algorithm 2: (1) Set NS ¼ {1, 2, . . . , N} and m ¼ 0. (2) Arbitrarily select a node i from NS, and use the depth-first search algorithm Tarjan (1972) to find the component CC(i) of graph G0 containing node i. Let m ¼ m þ 1 and NS ¼ NS\ NSCCðiÞ, where NSCC(i) denotes the node set of CC(i), NSCCðiÞ [ NSCCðiÞ ¼ f1, 2, . . . , Ng and NSCCðiÞ \ NSCCðiÞ ¼ ;. (3) Check the condition NS ¼ ;; if not, re-perform step (2); if so, go to step (4). (4) Arrange the m components of G0 in a sizedescending order. Then, relabel the nodes from the first component to the last one so as to obtain the graph G.
International Journal of Control Note that G0 and G are isomorphic to each other. And, under a given protocol (3), consensus in the closed-loop multi-agent system (2) with topology G0 can be achieved if and only if consensus in the closedloop multi-agent system (2) with topology G can be achieved. It is easy to check that the Laplacian matrix L of G is in the form of (1).
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In the following, consensus in multi-agent system (2) with topology G is discussed. Obviously, consensus cannot be achieved if G is disconnected. To guarantee consensus in system (2) with an arbitrarily given topology, a virtual leader labeled N þ 1 is introduced whose dynamics are given as follows: x_ Nþ1 ðtÞ ¼ AxNþ1 ðtÞ þ BuNþ1 ðtÞ, yNþ1 ðtÞ ¼ CxNþ1 ðtÞ,
ð23Þ
where xNþ1(t) 2 Rn is the state, uNþ1(t) 2 Rm is the control input, and yNþ1(t) 2 Rp is the measured output. Here, the virtual leader plays the role of a command generator providing a reference state for the followers to track. Thus, it is assumed that uNþ1(t) 0 and vNþ1(t) 0, i.e. the states of the virtual leader evolves without being effected by the followers, and the virtual leader is no need to observe the stats or outputs of any followers. It is furthermore assumed that only a subset of agents, called pinned agents, have access to the outputs of the virtual leader, but intermittently. For notational convenience, let P be the set of pinned agents and G be the augmented graph with adjacency matrix A ¼ ½aij ðNþ1ÞðNþ1Þ , where a(Nþ1)(Nþ1) ¼ 0, ai(Nþ1) 0, and ai(Nþ1) 4 0 if and only if i 2 P. The objective here is to find an appropriate control protocol for system (2) to achieve consensus in the sense of ki(t) Nþ1(t)k ¼ 0, 8i ¼ 1, 2, . . . , N, where Nþ1(t) ¼ (xNþ1(t)T, vNþ1(t)T)T. To do so, a pinningbased distributed protocol based on (3) is presented for each follower i as follows: v_i ðtÞ ¼ Avi ðtÞ þ Bui ðtÞ þ cF
N þ1 X
aij Cðvi ðtÞ vj ðtÞÞ
ðyi ðtÞ yj ðtÞÞ , ui ðtÞ ¼ Kvi ðtÞ, t 2 ½k!, k! þ Þ, v_i ðtÞ ¼ Avi ðtÞ, t 2 ½k! þ , ðk þ 1Þ!Þ,
Lemma 8: Suppose that for each i 2 {1, 2, . . . , m}, there exists at least one node ji such that ji 2 {V(i) \ P}. Then, matrix b L 4 0, where b L ¼ L þ , L is the Laplacian matrix of G, and ¼ diag{a1(Nþ1), a2(Nþ1), . . . , aN(Nþ1)} 2 RNN. Proof: Since, for each i 2 {1, 2, . . . , m}, there exists at least one node ji such that ji 2 {V(i) \ P}, it thus follows from Corollaries 1 and 2 in Ren, Beard, and McLain (2005) that L has a simple zero eigenvalue and all the other eigenvalues have positive real parts, where L is the Laplacian matrix of the augmented graph G. Taking M ¼ (a1(Nþ1), a2(Nþ1), . . . , aN(Nþ1))T 2 RN, one has " # b L M L¼ : ð25Þ 0T 0 where b L ¼ L þ . Thus, all the eigenvalues of b L have positive real parts. Since b L is symmetric, b L is positivedefinite. Based on the above analysis, an algorithm is given here to construct protocol (24). Algorithm 3: Suppose that the communication topology G0 has m component and that (A, B, C) is both stabilisable and detectable. Then, the consensus protocol (24) can be designed as follows. (1) Using Algorithm 2 to relabel the node indexes of G0 so as to get a graph G whose Laplacian matrix L has the form of (1). Then, pin m different nodes j1, j2, . . . , jm, where ji 2 V(i), and V(i) is the node set of the ith component of G, i ¼ 1, 2, . . . , m. (2) Choose & 0 4 0, and select a feedback gain matrix K by Ackermann’s formula, such that the real parts of the poles of A þ BK are smaller than & 0. (3) Solve the LMI AT Q þ QA 2CT C þ 2&0 Q 5 0
ð26Þ
to obtain one solution Q 4 0. Then, take the feedback gain matrix F ¼ Q1CT. (4) Take the coupling strength c 1/0, where 0 is the smallest eigenvalue of matrix b L.
j¼1
ui ðtÞ ¼ 0,
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To derive another main result, the following lemmas are introduced for which the proofs are simple therefore omitted. k 2 N,
ð24Þ
where vi(t) 2 Rn is the state of the observer embedded in agent i, c 4 0 is the coupling strength, and F 2 Rnp and K 2 Rmn are the feedback matrices. Before moving forward, the following lemma is introduced. Let V(i) be the node set of the ith (1 i m) connected component of G.
Lemma 9: The maximum real parts of the eigenvalues of matrices Ti are less than & 0, where A þ ci FC O Ti ¼ , ci FC A þ BK F ¼ Q1CT, Q is a solution of LMI (26), and i, i ¼ 1, 2, . . . , N, are the eigenvalues of b L.
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G. Wen et al. There exist some Wi 4 0 such that
Lemma 10:
Ti T Wi þ Wi Ti þ 2&Wi 5 0,
ð27Þ
where & 5 & 0, Ti ¼
A þ ci FC
O
ci FC
A þ BK
,
L. and i, i ¼ 1, 2, . . . , N, are the eigenvalues of b
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Now, it is to present another main result of this article. Theorem 2: For the multi-agent system (2) with a communication topology G0 , the protocol (24) constructed by Algorithm 3 solves the consensus problem if the communication rate /! 4 /(& þ ) þ ln%/ [(& þ )!], where 0 5 & 5 & 0, % ¼ maxi¼1, 2, . . . , N {max(Wi)/min(P), max(P)/min(Wi)}, 4 0, 0 is given in Lemma 7, and matrices P and Wi are positivedefinite solutions of LMIs (18) and (27), respectively. Proof: Take x i ðtÞ ¼ xi ðtÞ xNþ1 ðtÞ and vi ðtÞ ¼ vi ðtÞ vNþ1 ðtÞ, where i ¼ 1, 2, . . . , N. Obviously, consensus in (2) can be achieved if and only if kx i ðtÞk ! 0 and kvi ðtÞk ! 0, for all i ¼ 1, 2, . . . , N. Based on the above analysis and according to (24), one gets v_i ðtÞ ¼ Avi ðtÞ þ Bui ðtÞ þ cF
N X
b lij C vj ðtÞ x j ðtÞ ,
j¼1
ui ðtÞ ¼ Kvi ðtÞ, v_i ðtÞ ¼ Avi ðtÞ, ui ðtÞ ¼ 0,
t 2 ½k!, k! þ Þ,
t 2 ½k! þ , ðk þ 1Þ!Þ,
k 2 N,
ð28Þ h i where b L¼ b lij . NN T T Let i ðtÞ ¼ x i ðtÞ , vi ðtÞT , i ¼ 1, 2, . . . , N. Then, it follows from (2) and (3) that N P _i ðtÞ ¼ A1 i ðtÞ þ c b lij Hj ðtÞ,
t 2 ½k!, k! þ Þ,
j¼1
_i ðtÞ ¼ A2 i ðtÞ,
t 2 ½k! þ , ðk þ 1Þ!Þ,
k 2 N, ð29Þ
where
A BK A1 ¼ , O AþBK O O H¼ : FC FC
A A2 ¼ O
O , A
Similarly to the proof of Theorem 1, the rest of the proof can be completed. Remark 8: The minimum admissible communication rate for achieving consensus under a given
communication topology with fixed parameters & and can be obtained by minimising the parameter % in Theorem 2. And, the minimum % can be obtained by solving the following optimisation problem: (1) Minimise %i subject to: Wi 4 0, P 4 Wi, P 5 %iWi, AT4 P þ PA4 Ti T Wi þ Wi Ti þ 2&Wi 5 0, 2P 5 0, where A þ ci FC A O O , Ti ¼ , A4 ¼ O A ci FC A þ BK and i, i ¼ 1, 2, . . . , N, are the eigenvalues of b L. (2) Take %min ¼ maxi¼1, 2, . . . , N{%i}. Then, %min is the minimum value of %.
4. Simulation examples In this section, two simulation examples are provided to verify the theoretical analysis. Example 1: Take each agent in a multi-agent system to be a two-mass-spring system with a single force input (Zhang, Lewis, and Qu 2012), whose dynamics are described by (2) with 0 1 0 1 0 1 0 0 xi1 ðtÞ B x ðtÞ C B k1 k2 0 k2 0 C B i2 C B m C m1 xi ðtÞ ¼ B C, A ¼ B 1 C, 0 0 1A @ xi3 ðtÞ A @ 0 k2 2 0 k 0 xi4 ðtÞ m2 m2 0 1 0 B 1 C B C B ¼ B m1 C, ð30Þ @0A 0 where m1 and m2 are two masses, and k1 and k2 are spring constants. Furthermore, take the output matrix 0 1 1 0 0 0 B C C ¼ @ 0 1 0 0 A, 0 0 m1 ¼ 1.2 kg, k2 ¼ 1.0 N/m.
m2 ¼ 1.0 kg,
1
0 k1 ¼ 1.4 N/m,
and
Some simple calculations show that (A, B, C) is controllable and observable. Consider a group of four agents with undirected communication topology G1 as shown in Figure 1, where the weights are indicated on the edges. Obviously, G1 is connected. An observer-type of consensus protocol in the form of (3) is designed according to Algorithm 1. Take ! ¼ 4 and 0 ¼ 2.
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1
1.5
2
x11 (t) x21 (t) x31 (t) x41 (t)
1.0
0.4
4
3
xi1(t), i=1,2,3,4.
1 0.5 0 −0.5
Figure 1. Communication topology G1, in Example 1.
−1 −1.5 0
With communication
5
10 t
15
20
Without communication 3.2
3.2
4 x 12 (t) x 22 (t) x 32 (t) x 42 (t)
3 4
Figure 2. Intermittent communication.
Then, some calculations give the feedback gain matrices in (3) as K ¼ (27.12, 9.72, 3.32, 35.04) and 0 1 3:2838 0:6105 0:0594 B 0:6105 3:9534 0:6675 C B C ð31Þ F ¼B C: @ 0:0594 0:6675 8:1926 A 0:1739
0:0167
2
time
8
20:7378
According to Lemmas 6 and 7, one may take ¼ 1.15, ¼ 1.5. Letting c ¼ 2.75 and solving the optimisation problem in Remark 6 gives min ¼ 11. Thus, the minimum communication rate is 79.22%. Take ¼ 3.20, which means that the communication rate is 80%, see Figure 2 for illustration. According to Theorem 1, the states of all agents in system (2) will converge to the same value. The state trajectories of the agents are shown in Figures 3–6, respectively. Use vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 3 uX kxi ðtÞ x4 ðtÞk2 E1 ðtÞ ¼ t i¼1
xi2(t), i=1,2,3,4.
0
1 0 −1 −2 −3 −4 −5 −6 0
5
10 t
15
20
Figure 4. Consensus of trajectories of xi2(t), i ¼ 1, 2, 3, 4, in Example 1. 1.2 x13(t) x23(t) x33(t) x43(t)
1 0.8 xi3(t), i=1,2,3,4.
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Figure 3. Consensus of trajectories of xi1(t), i ¼ 1, 2, 3, 4, in Example 1.
0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
0
5
10 t
15
20
to denote the consensus errors of system (2) under protocol (3). Figure 7 demonstrates that consensus is indeed achieved.
Figure 5. Consensus of trajectories of xi3(t), i ¼ 1, 2, 3, 4, in Example 1.
Example 2: Consider a group of seven agents with a communication topology G2 as shown in Figure 8, where the agents 1, 2, 3 belong to the first component and agents 4, 5, 6, 7 belong to the second component. The agent dynamics and the parameters are the same
as those in Example 1. To achieve consensus, a virtual leader labeled by 8 is introduced. Then, an observertype of consensus protocol in the form of (24) is designed according to Algorithm 3. According to
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2 x 14(t) x 24(t) x 34(t) x 44(t)
xi1(t), i=1,2,...,8.
xi4(t), i=1,2,3,4.
1 0.5 0
−0.5 −1
1 0.5 0 −0.5 −1 −1.5 −2
0
5
10 t
15
20
Figure 6. Consensus of trajectories of xi4(t), i ¼ 1, 2, 3, 4, in Example 1.
0
5
10 t
15
20
Figure 9. Consensus of trajectories of xi1(t), i ¼ 1, 2, . . . , 8, in Example 2.
5 9
3
Consensus errors E1(t) xi2(t), i=1,2,...,8.
7 6 5 4
1 0 −1 −2 −3
2
−4
1
−5 0
5
10 t
Figure 7. Trajectories Example 1.
of
15
consensus
8
4
1
5
errors
E1(t),
in
1 1
1.2
1.75
2 4
3
0
5
10 t
15
20
20
5
7
6
2
3
0
x12 (t) x22 (t) x32 (t) x42 (t) x52 (t) x62 (t) x72 (t) x82 (t)
4
8
E1(t)
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x 11(t) x 21(t) x 31(t) x 41(t) x 51(t) x 61(t) x 71(t) x 81(t)
1.5
1.5
2
Figure 8. Communication topology G2, in Example 2.
step (1) of Algorithm 3, agents 1 and 7 are chosen as the pinned agents, i.e. P ¼ {1, 7}. Simple calculations yield that 0 ¼ 0.5493, where 0 is the smallest eigenvalue of b L given in (25). Take & ¼ 1.15, ¼ 1.5 and ! ¼ 4. Selecting c ¼ 2.75 and solving the optimization problem in Remark 8 gives %min ¼ 10.70. According to
Figure 10. Consensus of trajectories of xi2(t), i ¼ 1, 2, . . . , 8, in Example 2.
Theorem 2, the minimum communication rate is 78.96%. Take ¼ 3.20, which means that the communication rate is 80%, see Figure 2 for illustration. The feedback matrices F and K are taken to be the same as those in Example 1. Then, according to Theorem 2, the states of the agents in system (2) will approach those of the virtual leader. The state trajectories of agents are shown in Figures 9–12, respectively. Use vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 7 uX kxi ðtÞ x8 ðtÞk2 E2 ðtÞ ¼ t i¼1
to denote the consensus errors of system (2) under protocol (24). Figure 13 demonstrates that consensus is indeed achieved.
5. Conclusions In this article, the consensus problem for multi-agent systems with general linear node dynamics and
International Journal of Control 1.5
x13 (t) x23 (t) x33 (t) x43 (t) x53 (t) x63 (t) x73 (t) x83 (t)
xi3(t), i=1,2,...,8.
1 0.5 0 −0.5 −1 −1.5
0
5
10 t
15
20
Figure 11. Consensus of trajectories of xi3(t), i ¼ 1, 2, . . . , 8, in Example 2. x 14(t) x 24(t) x 34(t) x 44(t) x 54(t) x 64(t) x 74(t) x 84(t)
xi4(t), i=1,2,...,8.
1.5 1 0.5 0 −0.5
been proved that consensus in the closed-loop multiagent systems can be guaranteed under a fixed connected undirected topology if the communication rate is larger than a threshold value. Furthermore, consensus for multi-agent systems with a disconnected communication topology has been studied from a pinning-based distributed control approach. The effectiveness of the theoretical analysis has been verified by numerical simulations. Future work will focus on consensus for multi-agent systems with higher-order nonlinear dynamics and discontinuous output measurements.
Acknowledgements The authors sincerely thank the Associate Editor and all the anonymous reviewers for their valuable comments which helped them improve the manuscript of this paper. This work was supported by the Hong Kong Research Grants Council under the GRF Grant CityU 1114/11E, the National Nature Science Foundation of China under Grants 60974078, 90916003, 61104153, the China Postdoctoral Science Foundation under Grants 20100480211, 201104059 and the Information Processing and Automation Technology Prior Discipline of Zhejiang Province-Open Research Foundation under Grant 20120801.
−1 −1.5
0
5
10 t
15
References
20
Figure 12. Consensus of trajectories of xi4(t), i ¼ 1, 2, . . . , 8, in Example 2. 9 Consensus errors E2(t)
8 7 6 E2(t)
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2
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5 4 3 2 1 0
0
5
10 t
15
20
Figure 13. Trajectories of consensus errors E2(t), in Example 2.
discontinuous output measurements has been studied. To achieve consensus, a class of observer-type of protocols have been proposed. By using tools from switching systems theory and matrix analysis, it has
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