Consensus Problem in Multi-Agent Systems with Communication Channel Constraint on Signal Amplitude

SICE Journal of Control, Measurement, and System Integration, Vol 6, No 1, pp 007 013, January 2013 Consensus Problem in Multi-Agent Systems with Communication Channel Constraint on Signal Amplitude MingHui WANG and Kenko UCHIDA Abstract : The consensus problem of multi-agent systems with communication channel constraint is studied by providing special Laplacians of directed graphs in this paper Communication delays as one kind of channel constraints in multiagent systems have been discussed in the recent literature Different from those works, the consensus problem with communication channel constraint on signal amplitude is addressed in this paper Our work shows that the consensus can be obtained as long as some state-dependent switching parameters are introduced into two different types of the consensus protocols It is shown in an identical constraint case that the system given by one of the two protocols can reach globally asymptotically consensus and that the system can also achieve a generalized average-consensus if the directed graph is balanced; it is shown in a non-identical constraint case that the other protocol can provide the system with an asymptotic consensus Examples have been given to illustrate the effectiveness of the methods Key Words : consensus protocol, graph Laplacians, multi-agent systems, communication channel constraint 1 Introduction In recent years, as a new field of research, consensus problems of multi-agent systems have drawn substantial attentions from various fields such as vehicle formations, attitude alignment, rendezvous problem, flocking, and so on [1] [5] A multi-agent dynamic system, in general, can be described as a network of a number of loosely coupled dynamic units that are called agents Group cooperative behavior can be informally interpreted as giving consent to providing one s state and following a common protocol that serves the group objective [6] In networked multi-agent systems, agents may need to agree on some information which could be agent position, velocity, oscillation phase and decision variable Each agent makes decisions based on information about the state of the entire network that is obtained via communication with its immediate neighbors [7],[8] A critical problem for distributed coordinated control of multiple agents is to design appropriate protocols (or algorithms) such that the group of agents can reach consensus on the shared information [1],[9] Consensus problems have a long history in the field of computer science, particularly in automata theory and distributed computation [10] The theoretical framework for posing and solving consensus problems for networked dynamic systems was introduced by Olfati-Saber and Murray in [11] and [12] building on the earlier work of Fax and Murray [1] The study of the alignment problem involving reaching an agreement - without computing any objective functions - appeared in the work of Jadbabaie et al [5] Olfati-Saber and Murray discussed three consensus problems ie, directed networks with fixed topology, directed networks with switching topology, Department of Electrical Engineering and Bioscience, Waseda University, Tokyo 169-8555, Japan Department of Electrical Engineering and Bioscience, Waseda University, JST CREST, Tokyo 169-8555, Japan E-mail: kuchida@uchielecwasedaacjp (Received May 8, 2012) (Revised July 27, 2012) undirected networks with communication time-delay and fixed topology in [12] Further theoretical extensions of this work were presented in [13] [22] Beard and Stepanyan have defined notions of global asymptotic synchronizability and have shown that a group of agents is globally asymptotically synchronizable if and only if there exists a spanning tree of the communication graph [13] Ren and Beard have considered the problem of information consensus under dynamically changing interaction topologies and weighting factors [17] Jin and Zheng provide with some new inspection to study the asymptotically collective behavior of a multi-agent system under directed networks mainly by means of matrix analysis and graph theoretical approach [19] Munz et al have designed controllers that achieve consensus for a class of nonlinear multi-agent systems with relative degree two via networks with and without communication constraints in [21] and have showed that consensus in singleintegrator multi-agent systems is robust to arbitrary large delays in [22] In real problems, it is very important to take into account channel constraints such as time-delay As a result, research on time-delay systems and their control has been active in the last decade However, the problem about the communication channel constraint on signal amplitude has seldom been discussed The purpose of this paper is to explore conditions for the consensus problem of multi-agent systems with communication channel constraint on signal amplitude This paper discusses two types of Laplacians of network topologies in multiagent systems Then the consensus convergence criterion of systems is proposed Finally, some examples of three agents verify the rightness of the theoretics The two types of Laplacians introduced in this paper define two graphs with specific network topologies that may change as the states of the agents proceed From the state dependence of network topology, we could say that the two graphs are some special types of general dynamic graphs However, to the best of the authors knowledge, the proposed types of dynamic JCMSI 0001/13/0601 0007 c 2012 SICE

8 SICE JCMSI, Vol 6, No 1, January 2013 graphs have not been discussed so far; in the literature of dynamic graphs (see, eg, [23] and references therein), most research considers the case when the network topology depends on the relative states, while the network topologies in this paper depend on the absolute states, which comes from the communication channel constraint on absolute value of the transmitted state signal The rest of the paper is organized as follows In section 2, preliminaries about algebraic graph theory and the multi-agent systems are put forward In section 3, some concepts about the communication channel constraint are established In section 4, two consensus protocols and main results are stated In section 5, examples and simulation results are given for illustration Finally, concluding remarks are stated in section 6 The material in this paper was partially presented at the 2012 SICE Annual Conference [24] 2 Preliminaries In this section, we introduce some basic concepts of consensus and notations of algebraic graph theory that are often used in consensus problems of multi-agent systems and related to our later discussion More details can be found in [23],[25] Let G = (V, E, A) be a weighted directed graph of order n which consists of a vertex set V = {v 1, v 2,, v n }, an edge set E V V, and a weighted adjacency matrix A = [a ij ] R n n with a ii = 0 There is an edge of G which is denoted by e ij = (v i, v j ) if and only if the vertex v i receives information from the vertex v j Then the adjacency elements corresponding to the edges of the directed graph are nonzero A graph is called complete if every pair of vertices are adjacent If there is a path between any two vertices of a graph, then this graph is connected, otherwise disconnected A directed graph is strongly connected if any two vertices can be joined by a path A spanning subgraph with no cycles is called a spanning tree We see that a graph has a spanning tree if and only if it is connected Let I = {1, 2,,n} represents a set of cooperative agents with the total number n Assume that the communications among these agents are directed Then, we define as G = (V, E) as a directed graph where the n vertices represent n agents labeled as 1, 2,,n The agent i receives the information of its neighbor agent j, if there is a link (i, j) connecting the two vertices The information states with agent dynamics are given by ẋ i = u i, i = 1, 2,,n, (1) where x i R denotes the information state of the ith agent and u i R is the control input A consensus protocol to reach a consensus with respect to the states of n integrator agents (1) can be expressed as u i = a ij (x i (t) x j (t)), x i (0) R, (2) where a ij is the (i, j)th entry of the adjacency matrix of the associated communication graph, and N i represents the set of agents whose information is available to the agent i By applying the protocol (2), we can rewrite (1) into a firstorder linear system on a graph, ẋ(t) = Lx(t), (3) where x = (x 1, x 2,, x n ) T and L = [l ij ] R n n is the graph Laplacian of the network and its elements are defined as follows: n k=1 l ij = a ik, j = i a ij, j i Definition 1 [12] We say the vertex v i of a digraph G = (V, E, A) is balanced if and only if its in-degree and out-degree are equal, ie deg out (v i ) = deg in (v i ) A graph G = (V, E, A) is called balanced if and only if all of its vertices are balanced, or a ij = a ji i j j It is said that vertex v i and the vertex v j reach a consensus in a network if x i = x j We say the network has reached a consensus if x i = x j for all i, j I, i j Whenever the states of a network are all in consensus, the common value of all vertices is called the group decision value or consensus value Based on the system (3), we can get the following definition of the consensus protocol Definition 2 [13] The set of agents I is said to reach global consensus asymptotically if for any x i (0), i = 1, 2,,n, x i (t) x j (t) 0ast for each (i, j), i, j = 1, 2,,n Particularly, if as t, x i (t) Ave(x(0)) = ( n x i (0))/n, we say the system (2) achieves average consensus asymptotically The symbol denotes the Euclidean norm Lemma 1 [18] Suppose that L = [l ij ] R n n satisfies that l ij 0, i j, nj=1 l ij = 0, i = 1, 2,,n, and denote 1 n = (1,,1) T R n Then the following five conditions are equivalent, (i) L has a simple zero eigenvalue with an associated 1 n and all other eigenvalues have positive real parts; (ii) Lx = 0, x = (x 1, x 2,, x n ) T implies x 1 = x 2 = = x n ; (iii) Global consensus is reached asymptotically for the system (3); (iv) The directed graph with L as the Laplacian has a directed spanning tree; (v) rank(l) = n 1 Theorem 1 [6] Suppose G is a strongly connected digraph Then, (i) Global consensus is asymptotically reached for the system (3); (ii) If the digraph is balanced, an average-consensus is asymptotically reached Definition 3 [26] A real n n matrix M is a Metzler matrix if m ij 0foralli j In other words, M is a Metzler matrix if all nondiagonal elements are nonnegative Definition 4 [14] Consider an n n Metzler matrix M with zero row sums The δ digraph (δ 0) associated to M is a digraph with the vertex set {1, 2,,n} and with an arc from l to k (k l) if and only if the element of M on the kth row and the lth column is strictly larger than δ Theorem 2 [14] Consider the linear system ẋ(t) = L(t)x(t) Assume that the system matrix L(t) is a bounded and piecewise continuous function of time Assume that, for every time t, the system matrix is Metzler with zero row sums If there is an

SICE JCMSI, Vol 6, No 1, January 2013 9 index k {1,,n}, a threshold value δ>0 and an interval length T > 0 such that for all t R the δ digraph associated to t+t t L(s)ds, has the property that all vertices may be reached from the vertex k, then the equilibrium set of consensus states is uniformly exponentially stable In particular, all components of any solution x(t) of the system converge to a common value as t Remark 1 In the proof of Theorem 2 in [14], we have the following results on the condition of Theorem 2: x max (t) = max{x 1 (t),,x n (t)} is a non-increasing function, and x min (t) = min{x 1 (t),,x n (t)} is a non-decreasing function These two results are the foundation of our proofs later 3 Consensus Problem with Communication Channel Constraint In this section, we introduce some concepts about the communication channel constraint in multi-agent systems The information states with agent dynamics are given by ẋ i = u i, x i (0) R, i = 1, 2,,n, (4) where x i R denotes the information state of the ith agent and u i R is the control input As to the information acquisition system for each agent, we assume that the agent i has a sensor system to identify its own information state x i, and receives output signals y ij from the agent j through communication channel with constraints on signal amplitude which are described as follows: x j, x j b ij y ij =, i, j = 1, 2,,n, (5) φ, x j > b ij where φ denotes the agent i receives no information from the agent j, andb ij is the amplitude constraint parameter of the communication channel from the agent j to the agent i Of course, those constraints are caused by physical conditions of communication channels If x j denotes the velocity of the motion of the agent j, the channel constraint (5) implies that the agent i doesn t take into account the velocity x j faster than b ij A consensus protocol to reach a consensus with respect to the states of n integrator agents (4) can be expressed as u i = a ij (y ij )(x i y ij ), x i (0) R, (6) where a ij is the (i, j)th entry of the adjacency matrix of the associated communication graph at time t, andn i represents the set of agents whose information is available to the agent i Note a ij may depend on y ij Remark 2 Note that the communication channel constraint (5) is the condition of the absolute states In the research area on dynamic graphs (see, eg, [23]), the consensus problem under the constraint x i x j b ij, which is the condition of the relative states to represent the sensor range limit, is solved successfully, but does not cover our problem under the constraint (5) as a special case 4 Two Consensus Protocols In this section, we present two consensus protocols that solve consensus problems in a network of continuous-time integrator agents The first linear consensus protocol is defined as u i = a ij (x i σ ij x j ), (a ij > 0), x i (0) R, (7) and the following is the second linear consensus protocol: u i = σ ij a ij (x i x j ), (a ij > 0), x i (0) R, (8) where σ ij is defined as follows: 1, y ij = x j ( x j b ij ) σ ij = 0, y ij = φ( x j > b ij ), where b ij is the communication channel constraint from the agent j to the agent i To make the difference between the protocol (7) and the protocol (8) clear, we present an example: Consider a case of three agents I = {1, 2, 3} and focus on the control input of the agent 1 to show the difference between control structures given by the protocol (7) and the protocol (8) If the agent 1 receives the information y 12 = x 2 and y 13 = x 3 (ie, x 2 b 12 and x 3 b 13 ), the protocol (7) and the protocol (8) provides the same control u 1 = a 12 (x 1 x 2 ) a 13 (x 1 x 3 ); if the agent 1 receives only the information y 12 = x 2 (ie, x 2 b 12 and x 3 > b 13 ), the protocol (7) provides the control input u 1 = a 12 (x 1 x 2 ) a 13 x 1 and the protocol (8) provides the control input u 1 = a 12 (x 1 x 2 ); if the agent does not receive any information from other agents (ie, x 2 > b 12 and x 3 > b 13 ), the protocol (7) provides u 1 = (a 12 + a 13 )x 1 and the protocol (8) provides u 1 = 0 Remark 3 If for all i, j, b ij =, it is obvious that for all i, j,σ ij = 1 and at this time the consensus protocol (6) falls into the form of the consensus protocol (2) Then it follows form Theorem 1 that the consensus protocol (6) can be asymptotically reached for all initial states By applying the protocol (7) or (8), we can rewrite (4) into ẋ(t) = L σ(t) x(t), (9) where L σ is defined as the following L σ 1 and Lσ 2, for the consensus protocol (7) and (8) respectively j a 1 j σ 12 a 12 σ 1n a 1n σ 21 a 21 j a 2 j σ 2n a 2n L σ 1 = L σ 2 = σ n1 a n1 σ n2 a n2 j a nj j σ 1 ja 1 j σ 12 a 12 σ 1n a 1n σ 21 a 21 j σ 2 ja 2 j σ 2n a 2n, σ n1 a n1 σ n2 a n2 j σ nja nj where a ij > 0foralli, j Next are the main results that we present in this section Lemma 2 The Laplacian matrix is assumed to be L σ 2 The Laplacian matrix L σ 2 has the following properties: (1) If for all i, j, σ ij = 1, then L σ 2 1 n = 0; (2) If there exists j, such that for all i, σ ij = 1, then L σ 2 has a simple zero eigenvalue and all non-zero eigenvalues of L σ 2 have negative real parts,

10 SICE JCMSI, Vol 6, No 1, January 2013 Proof (1) It is obvious from the definition of L σ and Lemma 1 (2) If there exists j, such that for all i, σ ij = 1, then it follows from a ij > 0foralli, j that, for the digraph G σ with L σ 2 as its Laplacian, there should be always a edge from the vertex i (i j) tothevertex j Then we can get a directed spanning tree with the root vertex j According to Lemma 1, L σ 2 has a simple zero eigenvalue and all non-zero eigenvalues of L σ 2 have negative real parts Now we can state the main results: Theorem 3 for the protocol (7) and Theorem 4 for the protocol (8) Theorem 3 Consider a network of integrators with a fixed topology G = (V, E, A) that is a complete digraph and satisfies j a ij > 0foralli; constraint parameters [b ij ] of the communication channels in the network are identical such as b ij = b for all i, j Then, (i) There exists a time t c > 0, such that σ ij (t c ) = 1, i, j = 1, 2,,n, and the system (9) given by (7) can solve the global consensus problem asymptotically (ii) The protocol (7) globally asymptotically achieves the following average-consensus: x(t) 1 n ( n x i (t c )), as t,if G is balanced Proof Let the set I 1 denote the subset of the cooperative agents set I whose elements initial states are less than or equal to the identical communication channel constraint b ThesetI 2 is the complementary set of the set I 1 in I, that is, the elements initial states of the set I 2 are greater than the identical communication channel constraint b It is known that if the agent j belongs to I 1, there exists c such that x j c < b and σ ij = 1foralli We suppose that the agent i belongs to I 2, ie, x i > b ji = b and σ ji = 0forall j ẋ i = a ij (x i σ ij x j ) = a ij x i + a ij σ ij x j j I 1 + a ij σ ij x j j I 2 (σ ij = 1 for j I 1 ) = a ij x i + a ij 1 x j j I 1 + a ij 0 x j j I 2 (σ ij = 0 for j I 2 ) a ij x i + a ij 1 c j I 1 + a ij 1 c j I 2 (a ij > 0 and c > 0) = a ij (x i c) Then we also have d dt (x i c) a ij (x i c) By the Gronwall s inequality, we have x i (t) c (x i (t 0 ) c)e a ij (t t 0 ), x i (t) c + (x i (t 0 ) c)e a ij (t t 0 ) (10) The above evaluation of (10) assures the existence of a finite time t 1 (t 1 t 0 ), such that x i (t 1 ) = b Since then, we know i I 2 and x i (t 1 )( x i (t 1 ) = b) becomes a new initial state of the agent i Further, using again the evaluation of (10), we have that at the time t 1 +ε (ε is a small enough positive number), x i (t 1 +ε) < b holds, because on the closed interval [t 0, t 1 +ε], there is no agent moving from I 1 to I 2 in accordance with Remark 1 Similarly, for other elements in I 2, there also exist t 2, t 3,, such that all other elements in I 2 fall into I 1 Eventually, there must exist a time t c such that for any time t t c, all agents belong to I 1 and σ ij (t c ) = 1foralli, j Atthattime,L σ 1 turns into L as in the usual case, that is, the system (9) is reduced to the system (3); at this time, x(t c ) as a new initial state and the protocol (7) globally asymptotically solves the consensus problem according to Theorem 1 Thus, the proof of (i) is obtained Because the directed complete graph G is a strongly connected and balanced graph, the proof of (ii) can be obviously obtained from Theorem 1 Remark 4 Clearly 1 n ( n x i (t c )) b If we generalize the signal amplitude constraint x i (t) b to x i (t) x b for x R n,thenwehave 1 n ( n x i (t c )) x b Thus, b and x can be used as design parameters for specifying the consensus value Theorem 4 Consider a network of integrators with a fixed topology G = (V, E, A) that is a complete digraph; constraint parameters [b ij ] of the communication channels in the network are specified as b ij = b j for all i Then, the system (9) given by the consensus protocol (8) reaches consensus asymptotically for any initial state x(0) that has at least one element, say x j (0), satisfying x j (0) b j Proof Let {σ} be the set of all 2 n n matrices σ = [σ ij ] R n n where σ ij {0, 1} Denote by {σ} S 2 the subset of {σ} whose element σ makes L σ 2 have a simple zero eigenvalue and negative real-part eigenvalues Let x(t), t 0 be a trajectory of the system (9) with the protocol (8) and any given initial state x(0) satisfying x j (0) b j We first show that σ(t) {σ} S 2 for each t along the trajectory (Step 1), and prove that the system (9) with this switching parameter σ(t) achieves asymptoticconsensus (Step 2) In the following, we use a simplified notation σ ij = σ j for all i, which is justified by the assumption b ij = b j for all i (Step 1) First note that x j (t) b j implies σ j (t) = 1 at each time t 0 Hence, σ(0) {σ} S 2 follows from the assumption x j (0) b j and Lemma 2, and further it follows from Lemma 2 that σ(t) {σ} S 2 for any time t > 0 as long as x j (t) b j holds If there exist a time t a 0 and a small number ε >0 such that x j (t a ) = b j and x j (t a + ε) > b j for all ε in (0, ε], where x j (t a + ε) x j (t a ) ta +ε = σ k(τ)a jk [x j (τ) x k (τ)]dτ, (11) k t a then the identity (11) enables us to find a subscript k where k j and a small number η (0 < η ε) such that σ k (t a + η) = 1 for all η in (0, η]; thus, in this case, Lemma 2 again assures that σ(t a + η) {σ} S 2 for all η in (0, η](contrary, if x j (t a + ε) b j for all ε>0, it is a matter of course that σ(t a + ε) {σ} S 2 for all ε>0) Repeating the above argument, we can conclude that σ(t) {σ} S 2 for all t 0 along all the trajectory satisfying the assumption on the initial state x j (0) b j

SICE JCMSI, Vol 6, No 1, January 2013 11 (Step 2) From Step 1, for the system (9) given by the consensus protocol (8), we know that the Laplacian L σ(t) 2 is Metzler with zero row sums and the digraph with L σ(t) 2 as its Laplacian always has a directed spanning tree From what is stated above, the conditions of Theorem 2 are satisfied and then following from Theorem 2 the global consensus can be reached asymptotically Remark 5 In Theorem 4, assume that σ ij takes 0 or d (d is a positive number instead of 1), i, j = 1, 2,,n, the consensus can be more quickly reached asymptotically if d > 1 (see Example 4), or be more slowly reached asymptotically if 0 < d < 1 5 Examples and Simulation Results This section presents some illustrative examples to describe the theoretical results in this paper The following directed graph with different weights is needed in the analysis of this section Example 1 Figure 1 is a complete digraph with order n = 3 I = {1, 2, 3} represents cooperative agents at the vertices By simulation study, we investigate the consensus convergence character of the multi-agent systems and verify the proposed Theorem 3 in this example We suppose the initial states are x(0) = (36, 16, 2) T The identical constraint of communication channel is [b ij = b = 2] Based on Theorem 3, the consensus can be reached with x(t c ) as a new initial state (t c = 0317) and the simulation of these three agents is in Fig 2 The usual consensus [b ij = b = ] is shown in Fig 3 Example 2 We consider the system which is the same as that stated in Example 1 except for having a balanced complete digraph with a ij = 2foralli, j = 1, 2, 3 (i j) By simulation study, we investigate the generalized average-consensus convergence character of the multi-agent systems and verify the second statement proposed in Theorem 3 By Theorem 1, a usual average consensus is reached as shown in Fig 4 By Theorem 3, the consensus can be reached with x(t c )as a new initial state (t c = 0294) and the simulation of these three agents is shown in Fig 5 Furthermore, we notice that x(t c ) = (2, 0966, 1034) T and the consensus value is equal to 1 3 3 x i(t c ) = 13333, which is what we call the generalized average consensus Example 3 We suppose the initial states of three agents are x(0) = (36, 24, 16) T in Fig 1 The identical constraint of communication channel is [b ij = b = 2] Based on Theorem 4, the consensus can be reached with x(t a ) as a new initial state (t a = 09722) and the simulation of these three agents is shown in Fig 6 The usual consensus is like in Fig 7 Example 4 In Example 3, we took σ ij = 0or1andherein this example we assume σ ij = 0 or 5 as stated in Remark 5 and by Theorem 4, the consensus can be reached with x(t a )asa new initial state (t a = 01944) and the simulation of these three agents is shown in Fig 8 It can be seen from Fig 8 and Fig 6 that the consensus can be more quickly reached asymptotically in Fig 8 Furthermore, we notice that the time t a infig6is five times as great as another one in Fig 8 Fig 2 The consensus of three agents based on Theorem 3 Fig 3 Fig 4 The usual consensus of these three agents The usual average consensus of three agents Fig 1 The communication topology of three agents Fig 5 The generalized average consensus of three agents

12 SICE JCMSI, Vol 6, No 1, January 2013 of these three agents is shown in Fig 9 Note the trajectory of the first agent x 1 coincides with the top line, and the value of x 1 is equal to the consensus value Fig 6 The consensus of three agents based on Theorem 4 Fig 7 The usual consensus of three agents in Example 3 Fig 8 The consensus of three agents based on Remark 5 Fig 9 The consensus of three agents in Example 5 Example 5 In this example, we consider the case that the multi-agent systems have different communication channel constraints for different agents For example, the constraint of communication channel is b 21 = b 31 = 3andb 12 = b 32 = b 13 = b 23 = 15 The initial state of the multi-agent systems is x(0) = (26, 2, 17) T The topological structure of the multiagent systems is the same as Example 1 Finally, based on Theorem 4, the consensus can be reached and the simulation 6 Conclusion In this paper, the consensus problem in the multi-agent systems with the communication channel constraint on signal amplitude has been investigated by providing a special Laplacian representing the topological structure of the multi-agent systems Two types of the protocols using state-dependent switching parameters are introduced The present work shows that the protocols can obtain the global consensus as long as some conditions on the graph topology and the channel constraints are satisfied Examples have been presented to illustrate the effectiveness of the methods Further research will consider the consensus problem in multi-agents system with time-delays References [1] JA Fax and RM Murray: Information flow and cooperative control of vehicle formations, IEEE Trans Automatic Control, Vol 49, No 9, pp 1465 1476, 2004 [2] W Ren: Consensus based formation control strategies for multi-vehicle systems, Proc of the 2006 American Control Conference, pp 4237 4242, 2006 [3] R Olfati-Saber: Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans Automatic Control, Vol 51, No 3, pp 401 420, 2006 [4] JR Lawton and RW Beard: Synchronized multiple spacecraft rotations, Automatica, Vol 38, No 8, pp 1359 1364, 2002 [5] A Jadbabaie, J Lin, and AS Morse: Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans Automat Control, Vol 48, No 6, pp 988 1001, 2003 [6] R Olfati-Saber, JA Fax, and RM Murray: Consensus and cooperation in multi-agent networked systems, Proc of the IEEE, Vol 95, No 1, pp 215 233, 2007 [7] FX Tan, DR Liu, and XP Guan: Consensus value of multi-agent networked systems with time-delay, Proc of IEEE/INFORMS International Conference on Service Operations, Logistics and Informatics, pp 179 184, 2009 [8] W Ren, RW Beard, and EM Atkins: Information consensus in multivehicle cooperative control: Collective group behavior through local interaction, IEEE Control Systems Magazine, Vol 27, No 2, pp 71 82, 2007 [9] N Hovakimyan, E Lavretsk, BJ Yang, and AJ Calise: Coordinated decentralized adaptive output feedback control of interconnected systems, IEEE Trans Neural Networks, Vol 16, No 11, pp 185 194, 2005 [10] NA Lynch: Distributed Algorithms, Morgan Kaufmann Publishers, 1997 [11] R Olfati-Saber and RM Murray: Consensus protocols for networks of dynamic agents, Proc of the 2003 American Control Conference, pp 951 956, 2003 [12] R Olfati-Saber and RM Murray: Consensus problems in networks of agents with switching topology and time-delays, IEEE Trans Automatic Control, Vol 49, No 9, pp 1520 1533, 2004 [13] RW Beard and V Stepanyan: Information consensus in distributed multiple vehicle coordinated control, Proc of the 42th IEEE Conference on Decision and Control, pp 2029 2034, 2003 [14] L Moreau: Stability of continuous-time distributed consensus algorithms, Proc of the 43rd IEEE Conference on Decision Control, pp 3998 4003, 2004 [15] L Moreau: Stability of multi-agent systems with time-

SICE JCMSI, Vol 6, No 1, January 2013 13 dependent communication links, IEEE Trans Automatic Control, Vol 50, No 2, pp 169 182, 2005 [16] YG Sun and L Wang: Consensus of multi-agent systems in directed networks with nonuniform time-varying delays, IEEE Trans Automatic Control, Vol 54, No 7, pp 1607 1613, 2009 [17] W Ren and RW Beard: Consensus seeking in multi-agent systems under dynamically changing interaction topologies, IEEE Trans Automatic Control, Vol 50, No 5, pp 655 661, 2005 [18] W Ren and RW Beard: Distributed Consensus in Multi-vehicle Cooperative Control Theory and Applications, Springer, 2008 [19] J Jin and Y Zheng: Consensus of multi-agent system under directed network: A matrix analysis approach, Proc of the 7th IEEE International Conference on Control and Automation, pp 280 284, 2009 [20] YG Sun, L Wang, and GM Xie: Average consensus in networks of dynamic agents with switching topologies and multiple time-varying delays, Systems Control Letters, Vol 57, No 2, pp 175 183, 2008 [21] U Munz, A Papachristodoulou, and F Allgower: Robust consensus controller design for nonlinear relative degree two multi-agent systems with communication constraints, IEEE Trans Automatic Control, Vol 56, No 1, pp 145 151, 2011 [22] U Munz, A Papachristodoulou, and F Allgower: Consensus in multi-agent systems with coupling delays and switching topology, IEEE Trans Automatic Control, Vol 56, No 12, pp 2976 2982, 2011 [23] M Mesbahi and M Egerstedt: Graph Theoretic Methods in Multiagent Networks, Princeton University Press, 2010 [24] MH Wang and K Uchida: Consensus of multi-agent systems with signal amplitude constraint in communication, Proc of the 2012 SICE Annual Conference, pp 108 112, 2012 [25] C Godsil and G Royle: Algebraic Graph Theory, Springer, 2001 [26] DG Luenberger: Introduction to Dynamic Systems: Theory, Models and Applications, John Wiley and Sons, 1979 MingHui WANG (Student Member) He received his BS and MS degrees from Northeastern University, China, in 2007 and 2011, respectively He is currently a PhD student at Waseda University His research interests include linear control and multi-agent systems Kenko UCHIDA (Member, Fellow) He received the BS, MS and DrEng degrees of Electrical Engineering from Waseda University, Japan in 1971, 1973 and 1976, respectively He is currently a Professor in the Department of Electrical Engineering and Bioscience, Waseda University His research interests are in robust/optimization control and control problem in energy systems and biology He is a member of ISCE, IEEJ and IEEE