Abstract
It is well known that all agents in a multiagent system can asymptotically converge to a common value based on consensus protocols. Besides, the associated convergence rate depends on the magnitude of the smallest nonzero eigenvalue of Laplacian matrix . In this paper, we introduce a superposition system to superpose to the original system and study how to change the convergence rate without destroying the connectivity of undirected communication graphs. And we find the result if the eigenvector of eigenvalue has two identical entries , then the weight and existence of the edge do not affect the magnitude of , which is the argument of this paper. By taking advantage of the inequality of eigenvalues, conditions are derived to achieve the largest convergence rate with the largest delay margin, and, at the same time, the corresponding topology structure is characterized in detail. In addition, a method of constructing invalid algebraic connectivity weights is proposed to keep the convergence rate unchanged. Finally, simulations are given to demonstrate the effectiveness of the results.
1. Introduction
In recent years, many issues related to consensus have been studied [1â€“10]. This is due to the large number of potential applications of consensus, ranging from engineering and computer science to biology, ecology, and social science. In [1], Jadbabaie proposed a nearest neighbor law to coordinate the control of the whole system, so that agents can reach a common final value. OlfatiSaber and Murray proposed a simple consensus protocol to achieve average consensus for undirected graphs and balanced digraphs in [2]. Based on quasiconsensus, Cai proposed an approach for clustering in [3]. And Liu proposed several necessary and sufficient conditions for consensus of secondorder multiagent systems under directed topologies [4]. For highorder linear timeinvariant singular multiagent systems with constant time delays, admissible output consensus design problem was investigated in [5]. For hybrid multiagent systems, necessary and sufficient conditions were also developed for the consensus [6, 7].
The smallest nonzero eigenvalue of Laplacian is the algebraic connectivity, which was first proposed by Fiedler [11]. OlfatiSaber and Murray conducted a preliminary discussion on the convergence rate of consensus and proposed the concept of communication cost [2]. The algebraic connectivity increases as the communication cost goes up. An approach of minimizing the guaranteed cost was given in [12]. For a stable system, we always expect that the system has a larger convergence rate. Based on the consensus protocol, the convergence rate becomes larger as the algebraic connectivity of the system increases.
The convergence rate problem of consensus had been studied in [13â€“20]. The fast consensus and the optimization problem of convergence rate have been studied extensively [13]. The convergence rate of consensus was studied under the condition of weighted average in [14]. Jin proposed a multihop relay protocol to make a larger algebraic connectivity [15]. In [16], OlfatiSaber studied the fast consensus for smallworld networks through randomly rewiring edges. In [18], Yang studied the optimization problem of convergence rate for secondorder systems with time delays by the frequencydomain method. Yang proposed a new variable not only to measure the situation of optimization, but also to cope with the tradeoff between and . Xiao developed the results of delay margin by analyzing in [17]. An optimal consensus protocol minimizing team cost function was proposed in [19], and an optimal synchronization protocol was designed for the largest convergence rate and minimal steady state error when the protocol is perturbed by an additive noise in [20].
To improve the convergence rate of consensus in multiagent systems, we need to change the algebraic connectivity . The variation of algebraic connectivity depends on the variation of weights and topologies of communication graphs. The algebraic connectivity can be improved by taking advantage of algorithms and graph theory, which was explored in [21â€“24]. The convergence rate is susceptible to the perturbation, which will further affect the stability of the system [25â€“28]. Based on former researchersâ€™ works [29â€“37], we propose a concept of invalid algebraic connectivity weights (IACW), which is shown not only to be resistant to the perturbation but also can avoid unnecessary waste of costs. There are three methods to get a larger convergence rate: changing the protocol; changing the weights of a system; changing the topology of a system. In this paper, we take methods and . In particular, we investigate how the convergence rate varies when a superposition system is joined to the original system and present a detailed characterization for the variation of convergence rate. The results are developed by analyzing the variation of eigenvalues and eigenvectors of Laplacian matrix. Under fixed costs, we give the most optimal case, which can make the convergence rate and delay margin the largest. For a complete graph, if the cost is fixed, then reaches maximum and reaches minimum. Thus, the convergence rate and delay margin achieve the biggest magnitude. Since the topology is unique, this achieves the optimization of convergence rate and delay margin. In addition, the method and conditions are put forward for the construction of invalid algebraic connectivity weights.
In a multiagent network system, not all agents are cooperative. There are some agents which are competing. The cooperative and competitive relationship is represented, respectively, by positive weights and negative weights. Positive and negative weights have opposite effects on system performance. In some circumstances, negative weights are also needed. For example, in a system with time delay, the negative weights can reduce the magnitude of , which results in a larger delay margin. Hence, the fast consensus with antagonistic interactions is considered in the paper.
The rest of the paper is organized as follows. In Section 2, basic definitions, properties, and system model are addressed. In Section 3, we study the variation of convergence rate when a superposition system is superposed on the original system, and characterize the most optimal case of convergence rate under fixed cost. In Section 4, results are derived for the identification of invalid algebraic connectivity weights and the associated construction of IACW. In Section 5, simulation results are presented to show the effectiveness of the approach. Finally, conclusions are given in Section 6.
2. Consensus Protocol and Consensus State
Let be a weighted undirected graph, where denotes a set of nodes, denotes the set of edges, and is the adjacency matrix of undirected graph . In this paper, if has a communication link with , then there is an edge between nodes and , with , ; and is the weight between nodes and . Here, the selfloop of , under which , is not considered.
The firstorder multiagent system is given bywhere denotes the state of agent , which is the th component of ; denotes the set of adjacent agents of agent . The Laplacian matrix of is represented by , where is the diagonal connectivity degree matrix of . So (1) can be written asThe entries of can be written as can be written as
Assume that the communication graph consists of nodes, and the spectrum of eigenvalues is . The spectrum means that the eigenvalues are arranged according to a certain order. Let denote a vector space, in which any vector can be represented as a linear combination of ; i.e., . Below means that the entries of matrices and plus together. For , the entries of are , where , are the entries of , , respectively.
Definition 1. The system associated with graph , which has the same node set with the original graph , is called a superposition system. Let be superposed to . We get a connected graph . Then the system associated with is called a superposed system.
We give the definition of superposition system to represent the situation of the adding edges such as , numbers, and weights of edges, so that we study the variation of superposed systems under different situations of adding edges.
In this paper, graph is connected and the spectrums of eigenvalues associated with both and are identical with , where denotes the Laplacian matrix of , and denote the Laplacian matrix of . In what follows, means that is positive semidefinite. Although there are negative weights, can still be positive semidefinite or negative semidefinite.
3. Variation of Convergence Rate
The changing of algebraic connectivity relies on the variation of eigenvalues of Laplacian So the following lemma is introduced.
Lemma 2 (Weyl theorem [38]). Let be symmetric and the eigenvalues of and be ordered as that of , respectively. Thenfor each , with equality holding for some pair if and only if there is a nonzero vector such that . Also,for each , with equality holding for some pair if and only if there is a nonzero vector such that , , . If and share no common eigenvectors, then (5) and (6) are strict inequalities.
By Hermam Weyl theorem, some significant inequalities can be derived. This is essential to the development of results. In what follows, let , , .
3.1. Convergence Rate of Undirected Graphs
For a system which can achieve consensus, its eigenvalues satisfy and are arranged in an increasing order, . The solution of the system is , where is the initial state of agents. Let , where is the Jordan standard norm of , and the diagonal entries of are the eigenvalues of . Then . Let . It follows that if the multiplicity of all eigenvalues of is 1, where is the final value of the system. Therefore, the decay rate of the component in is less than or equal to the components of . If there is a repeated eigenvalue of , is a nonlinear generation of . Assume that the nonlinear generation of in is , . When , the derivative of its nonlinearity is less than or equal to the nonlinear generation corresponding to . Thus, the rate that converges to a steady final value is determined by .
The superposition system has the same node set with the original system, and its edges can be constructed arbitrarily. We aim to analyze the variations of convergence rate after joining different superposition systems. For a system with undirected graphs, the following theorem can be achieved after joining a superposition system.
Theorem 3. For an undirected connected graph , if is a positive semidefinite matrix, thenIf is negative semidefinite, thenOnly for with multiplicity 1, the necessary and sufficient condition for equalities in (7) and (8) to be true is that there is a vector such that , , .
Proof. Let ; ; ,,, be the eigenvectors of , , and , respectively, with , . Denote , , . Then which guarantees that there is a unit vector . So we obtain the following inequality:For , let be positive semidefinite; it follows that Denote , , , , . Then which guarantees that there is a unit vector . Thus,For , let be negative semidefinite; one has If is not a repeated eigenvalue, then only if there is an eigenvector associated with satisfying . In case that there is no eigenvector satisfying , then is larger than . So there are no vectors with == such that . Thus the necessary and sufficient condition for equalities in (7) and (8) to be true is that there is a vector satisfying , , and .
Assume that is a repeated eigenvalue with multiplicity If or changes and the other eigenvalues remain unchanged, and share eigenvalues, while the corresponding eigenvectors of are different from those of . Since has identical eigenvalues with , there is an eigenvector which has components satisfying , such that , , . In this case, however, the components of the corresponding eigenvector of are different from each other. That is, there are different vectors such that .
Theorem 3 shows that the convergence rate of a system can be increased or decreased by adding a superposition system. This is always true for the positive semidefinite matrix or negative semidefinite matrix , no matter what is the sign of the weights of . It is based on the analysis whether the equalities in (7) and (8) hold, we can figure out the variations of the convergence rate. In what follows, more specific instructions will be given on how to affect the convergence rate with respect to the problems and situations encountered during the process.
In (10), take , we see that . ThenIf is positive semidefinite, (15) holds. Take in (13). Then, based on (15), if is negative semidefinite, we have
Proposition 4. For an undirected connected graph , let represent the set of eigenvectors corresponding to with multiplicity , . If , and there exists a vector satisfying , then the multiplicity of corresponding to is .
Proof. Let be a positive semidefinite matrix. For in (13), and , we haveBy (15) and (17),Assume that there are repeated eigenvalues of Laplacian associated with graph , which are , , and there exists one vector satisfying . Note that not for every vector , holds. So there is one of repeated eigenvalues, the value of which is improved; i.e., , .
In case is negative semidefinite, the same line of arguments yields that , .
Proposition 4 shows if the eigenvector of one eigenvalue has two identical entries , adding an edge does not affect the eigenvalue , which means the weight, and even the existence of does not affect . For an eigenvalue with multiplicity , the corresponding eigenvector has eigenvectors with two identical entries at any positions, so that if , no matter the position of the adding edge, there is only one change; that is, the multiplicity of of superposed system is . It is easy to know that if the multiplicity of is , there is at least one eigenvector of that has pairs of identical entries; that is, if , the superposed system has the eigenvalue .
Example 5. The graph of the original system is shown as Figure 1(a). Introducing the superposition system only establishes a connection between nodes 1 and 2. Thus, . The graph of the superposed system is shown as Figure 1(b). Below are and The nonzero eigenvalues of the original system are , , , and , and the nonzero eigenvalues of the superposed system are , , , and . The multiplicity of eigenvalue 5 is 2. Let us join the superposition system with only one connection; i.e., . Since the eigenvector of the repeated eigenvalue satisfies , the multiplicity of is 1.
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3.2. Changes of Convergence Rate
Definition 6. For an entry of the adjacency matrix , , if the increase of its value does not affect the algebraic connectivity of the communication graph, we call these weighted entries invalid algebraic connectivity weights (IACW).
For an invalid algebraic connectivity weight , if there is a perturbation on one of associated nodes and so that the value of decreases, the magnitude of each eigenvalue except 0 and decreases firstly, while the delay margin of the system increases. Thus, in this case, the stability and the convergence of the system can be protected to a certain extent.
If there is an invalid algebraic connectivity weight in the system, then , where is an eigenvector of satisfying . All the eigenvector corresponding to the smallest nonzero eigenvalue must contain two components and with , , . Below is a further explanation.
Due to the existence of repeated eigenvalue, the convergence rate does not necessarily change when a superposition system is superposed to the original system.
Theorem 7. If a superposition system related undirected graph is superposed to an undirected connected graph , a composite connected graph is generated for the corresponding superposed system. Then the convergence rate of the superposed system changes as follows.(i)If there are invalid algebraic connectivity weights in , the adjacency matrix of the superposition system only consists of invalid algebraic connectivity weights, and is positive semidefinite, then the superposed system converges at a rate equal to the original system; i.e., .(ii)If and , where is positive semidefinite and is the eigenvector of , then the superposed system converges faster than the original system; i.e., .(iii)If , and , where is negative semidefinite and is the eigenvector of , then the superposed system converges slower than the original system; i.e., .
Proof. â€‰(i)If a system associated with the original graph has an invalid algebraic connectivity weight, then all the eigenvectors corresponding to the smallest nonzero eigenvalue of contain two components , with , , . only consists of invalid algebraic connectivity weights if the superposition system only establishes a connection between and , such that there is an eigenvector corresponding to with , . Therefore, the convergence rate of the superposed system remains unchanged.(ii)If and , where is positive semidefinite, then , the multiplicity of in is 0. By (7), holds.(iii)If and , where is negative semidefinite, then and the multiplicity of in is 0. By (8), and , which yields that is positive semidefinite.
Example 8. The graph associated with the original system is shown as Figure 2(e). The superposition system only establishes connections among nodes 3, 4, and 5. Thus, . The superposed system is shown as Figure 2(b). Superpose another superposition system to , where is shown as Figure 2(c) and only establishes the connection between nodes 1 and 2, which means . The superposed system is shown as Figure 2(d). , , and are given as follows. The nonzero eigenvalues of the original system are , , , and , and the nonzero eigenvalues of the superposed system are , , , and . The invalid algebraic connectivity weights are , , and . The established connections among nodes 1, 3, 4, and 5 mean . By Theorem 7, the convergence rate of the system still remains unchanged.
The nonzero eigenvalues of the superposed system are , , , and . Let the superposition system only establish a connection between nodes 1 and 2; that is, . Then, there is no eigenvector associated with satisfying . Therefore, the superposed system converges faster than the original one.
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Corollary 9. For an undirected connected graph with the multiplicity of being , let represent the multiplicity of of , where the Laplacian of the superposition system is positive semidefinite and , Thus, if , then , and accordingly the convergence rate of is larger than the original system.
Proof. Set , , where are different matrices, , , . If a superposition system is superposed to the original system, and there is an eigenvector of associated with satisfying , then . Let be an eigenvector corresponding to , . If does not hold for every , and there is no eigenvector corresponding to satisfying , then . Thus, if , then is not an eigenvalue of anymore. So , and accordingly the convergence rate of is larger than that of .
Remark 10. For any , if there is an eigenvector of such that , the multiplicity of eigenvalue associated with satisfies . That is to say, if , then , .
Because the same elements may exist in the eigenvector of , superposition systems do not always enhance the convergence rate. In case , the convergence rate changes only if the entries of associated with the edges of supposition system are not all equal. For example, if only the entries of are equal and the others are different from each other, then the superposition system with the connections among nodes 2, 3, 4, and 5 can enhance the convergence rate, while the superposition system with only the connections among nodes 3, 4, and 5 will not have such effect.
Algorithm 1 gives a method of finding a superposition system to enhance the convergence rate. In Algorithm, changes the label of node , which makes different nodes connect to node or nodes , until ; or until are not all equal. If , there exists at least one pair of entries with in the eigenvector corresponding to . If , there exists a set of elements in eigenvector that are not all equal; otherwise, is a zero vector. So there is a superposition system that can increase the rate of convergence.

Proposition 11. For an undirected connected graph , if the smallest nonzero eigenvalue of is repeated, the invalid algebraic connectivity weights can be constructed in any two nodes.
Proof. The invalid algebraic connectivity weights exist if and only if, for an eigenvector corresponding to the smallest nonzero eigenvalue of , there are two components of satisfying ,
For the positive semidefinite matrix or negative semidefinite matrix , let . If is a repeated eigenvalue, only one of repeated eigenvalues of will change. Then for the superposition system, the nonzero entries of or are invalid algebraic connectivity weights. All the eigenvectors corresponding to of contain components with , where represents the position of each nonzero weights associated with and .
3.3. Optimization of Convergence Rate
In a stable system, it is always expected that there is a larger convergence rate to take less time to get the stable state. In case all the weights of a communication graph are increased, the values of nonzero eigenvalues will be improved as a whole, and accordingly the stability can be achieved with a larger rate. From a practical point of view, however, this will increase the cost of realization. Even so, we still hope to achieve the rapidity of convergence under fixed cost. Below, let denote the cost for achieving consensus. With being fixed, we hope to find the most optimal topology to achieve the largest convergence rate.
Theorem 12. For a system with fixed cost , , if is timevarying and , then the system achieves the largest convergence rate.
Proof. Assume and the system does not achieve the largest convergence rate. If , then , which is in contradiction with the known condition. If , it can be seen that the system does not achieve the largest convergence rate. The above arguments mean that the system achieves the largest convergence rate if .
For the communication delay between nodes and , we consider the case Then system (2) along with protocol (1) can be written as
Lemma 13 (see [2]). Consider a network of integrator agents with identical communication time delay in all links. Assume that the network topology is fixed, undirected, and connected. Then, protocol (22) with globally asymptotically solves the average consensus problem if and only if either of the following equivalent conditions is satisfied.(i) with .(ii)The Nyquist plot of has a zero encirclement around .
By Theorem 12, is determined as long as the system achieves the largest convergence rate under fixed cost. Since the value of is determined, it is expected to construct topologies to achieve the largest convergence rate, i.e., seeking , where is the Laplacian having an eigenvalue . If the graph of a multiagent system is a complete graph and all of its weights are identical, then the system achieves the largest convergence rate under fixed cost , . In this case, achieves the biggest magnitude and achieves the smallest magnitude. By Lemma 13, the delay margin is decided only by . If there is a communication delay in the system and the achieves the smallest magnitude, then the delay margin achieves the biggest magnitude.
The Laplacian transformation of (22) isLet . Then . In case , we have . Let . The partial derivation of is . If is monotone for in , the root of can only be one side of . As a consequence, the rightmost root of a system with time delays is related to or .
Corollary 14. If , , and the communication topology is a complete graph with all weights being identical, then the delay margin achieves the biggest magnitude, and the topology is unique. And the system achieves largest convergence rate, if is monotone for .
Proof. If the communication graph associated with the system is a complete graph, each node of the graph is connected with the others, and all the weights are identical. Hence, , . Then achieves the biggest magnitude. Thus, achieves the smallest magnitude; that is, achieves the biggest magnitude.
If is monotone increasing for , then . Thus, the convergence rate is determined by the upper bound ; that is, , . increases with increasing. If is monotone reducing for , then . The convergence rate is determined by the lower bound ; that is, , . increases with reducing. If achieves the biggest magnitude, achieves the smallest magnitude, and is monotone for ; then the system achieves the largest convergence rate.
Remark 15. If a node is connected with all the other nodes, and the weights between and the others are identical, then is an eigenvalue of .
For a system with time delays , the convergence rate is also affected by time delays .
Corollary 16. For a fixed topology, if the time delay increases and , the convergence rate of the system reduces.
Proof. The characteristic polynomial of the system is , . Let , ; then the root of characteristic polynomial of the system can be solved from . Thus, , , , such that . , and then increases with increasing; that is, convergence rate reduces with increasing for .
Example 17. The original system is shown as Figure 3(a), and the system with the largest convergence rate under the same cost is shown as Figure 3(b). The corresponding Laplacian matrices are For the eigenvalues of original system, , , and the delay margin is . All the nonzero eigenvalues of the system with the largest convergence rate are 5, and the delay margin of this system is .
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4. Construction of Invalid Algebraic Connectivity Weights
Definition 18. is called an equitable weights partition if each node in cell has the same weight with the nodes in , . If each node in has the same weight with the nodes in the partition is called an almost equitable weights partition, where . We denote the cardinality of cell with .
A cell is nontrivial if it contains more than one node; otherwise it is trivial. Equitable weights partition can be used in what follows to analyze the convergence rate of consensus, and we can construct the nontrivial cells to get the IACWs.
4.1. Convergence Rate under Equitable Weights Partition
For a nontrivial cell in an equitable weights partition, suppose that the number of nodes in is . We have the following lemma.
Lemma 19. If are superposition systems only constructed, respectively, by and and and ; then , where , and is a node given in advance, .
Proof. Since there is a nontrivial cell with nodes and is fixed, we construct the superposition system by selecting any node in the nontrivial cell to establish one communication link with the given . The dynamic equations of the nodes in the nontrivial cell arewhere the entries of the columns in and are all equal. In the superposition system, an arbitrary node is selected to establish one communication link with the fixed . Then the dynamic equations remain unchanged when the positions of are exchanged, and the positions of remain unchanged. In this situation, the topology of the system remains unchanged. The discussion above means that the variations of eigenvalues associated with are equal since there is only one link between and any node of in the superposition system. Note that the variation of algebraic connectivity is also identical. Then, if , the variation of the algebraic connectivity is unrelated to the choosing of . Thus, if is selected to construct a superposition system, its convergence rate is identical with , and accordingly , where .
Example 20. A graph with two nontrivial cells is shown as Figure 4. The corresponding Laplacian isThe expression of Laplacian implies that there is a partition , , . Consider the case that , which means that there is an edge with weight in the graph associated with the superposition system. Let us establish the connection between any two nodes, and let denote the corresponding matrix which has only one link between nodes and . denotes the Laplacian matrix of the superposed system corresponding to . Then, , , . The variations of the other eigenvalues of are the same.
Let denote an eigenvalue of a submatrix which is generated by selecting the rows and columns corresponding to the nodes of the nontrivial cell . , with indicating that only relies on The denotes the vector which only contains two nonzero entries, and the two entries are opposite from each other.
Corollary 21. For an equitable weights partition with a nontrivial cell , the multiplicity of satisfies , , .
Proof. Since Laplacian is symmetric, the geometrical and algebraic multiplicity of each eigenvalue of is equal to each other. Hence, the number of linearly independent eigenvectors of is the multiplicity of . Note that is equal to the number of linearly independent solutions of . It follows that is equal to the number of linearly independent vectors which satisfies , . The structure form of implies that the row entries of corresponding to the nodes in are equal. Therefore, .
Remark 22. The number of eigenvalues is no less than the number of the nontrivial cells .
It is worth mentioning that, for an equitable weights partition, the submatrix corresponding to a nontrivial cell consisting of nodes only has the eigenvalue with its multiplicity satisfying . There, however, will be such a situation , which is the consequence of the nontrivial cell working with the other cells.
Proposition 23. For an equitable weights partit