ON THE ROLE OF MATRIX-WEIGHTS ELEMENTS IN CONSENSUS ALGORITHMS FOR MULTI-AGENT SYSTEMS

Copyright: © 2021 by the authors. Submitted to Network for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/ 4.0/). 1 Division of Electrical and Computer Engineering, School of Electrical Engineering and Computer Science, Louisiana State University; jogbeb1@lsu.edu; xmeng5@lsu.edu * Correspondence:xmeng5@lsu.edu ‡ These authors contributed equally to this work.


Introduction
Multi-agent systems (MASs) consists of multiple autonomous agents [1], which can 14 be used to solve problems that are difficult or impossible for an individual agent or 15 a monolithic system. Communication and interaction between individual agents are 16 fundamental characteristics of MASs, which allow agents to achieve a global objective 17 despite of having access to only local neighbourhood information [2], or edge informa- 18 tion [3,4]. Having multiple agents could speed up a system's operation by providing 19 a method for parallel computation, offering advantages in terms of extensibility and 20 flexibility compared to single agent systems [5]. Applications of MASs can be found in 21 cloud computing and social networks [6]. 22 Consensus is one of the fundamental problems in multi-agent coordination, where 23 agents interact with their neighbors according to a local protocol to ensure that a common 24 value in terms of the state components, is agreed upon globally, by all the agents. This 25 sweeping form of consensus is generally referred to as a global consensus. A clustered 26 consensus is also possible, where some agents agree on some values, different on the 27 consensus value of some other agents [7]. MAS is widely studied using the graph theory, 28 in which the vertices and edges represent agents and the inter-agent links, respectively. 29 Conventionally, the inter-agent links have been modelled by scalar weights [8,9]. The 30 consensus algorithm is widely applied in the agent level [10][11][12]. Matrix-weights can be 31 used to capture the complexity in the state level for MASs. Particularly, the consensus of 32 all agents in corresponding states may be affected by other states. There are a number of 33 works in the literature that extend the conventional scalar weighted graph to the matrix 34 weighted graph for the consensus problem. One of such is [13,14] where the conditions 35 for achieving consensus in matrix-weighted Discrete and continuous-time consensus 36 algorithms are presented using the properties of the graph Laplacian. In [14], discrete-37 time matrix-weighted consensus is studied over undirected and connected graphs 38 considering symmetric matrix weights and a special case of non-symmetric matrix-39 weights that can achieve consensus control. Lyapunov stability theory for discrete-time 40 systems is also employed to show the system's convergence to consensus. In the same 41 way, in [13], matrix-weighted consensus algorithm is studied with fixed undirected 42 graphs and a necessary and sufficient condition for exponentially reaching a global  When the agent-to-agent link is weighted by positive semi-definite (PSD) matrices, 53 it can be difficult to tell what form of consensus will be present. Clustered consensus can 54 happen even when the graph is connected in the agent level, due to the existence of PSD 55 matrix weights. In a case, when all the elements of each of the matrix-weights are set 56 to one, for a complete graph on five vertices, there will be no form of consensus for the 57 agents in any of the states.

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In this paper, we study how the choice of matrix weights for the inter-agent links 59 affect the presence of clusters in a matrix-weighted graph. A linear algebraic approach is 60 used in [15,16] to examine the properties of the Laplacian matrix and the necessary and 61 sufficient conditions are given for the existence of consensus. We adopt a matrix-element-62 mapping approach to study how the matrix-weights alter the consensus dynamics. Our where x i (t) ∈ R m and u i (t) ∈ R m denote the states and the inputs of agent v i , re- j of agent j has no influence on x [k] i of agent i. If there is at least one entry in W ij in which w ij kl > 0, we say agent v j is a neighbor of agent v i and v j v i ∈ E . The set of neighbors of agent v i is defined as arborescence, then we say the k-state graph is connected.

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The degree or valency of a given vertex d(v i ) ∈ R m×m is defined as that is, the sum of weights on the link from each neighbouring node for all states of as the block diagonal matrix of the degrees of all agents in G. The nm × nm adjacency The weighted Laplacian matrix of the graph, L W (G) ∈ R nm×nm is given by The with I m ∈ R m×m being the identity matrix, ∆ W (G), the nm × nm degree matrix of G, is the 96 diagonal matrix of the degree of all the vertices in G given by and the nm × nm adjacency matrix A W (G) is defined as The matrix-weighted Laplacian of the graph, L W (G) ∈ R nm×nm is given by i and x [k] j satisfy the condition then we say they belong to the same cluster. All k-

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It is easy to verify that all k-clusters satisfy the conditions

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Note that for different entries i and j of all states, the i-clusters C i and the j-clusters 112 C j may not be the same, that is,

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Our main aim in this paper is to examine how the elements of the matrix-weights 121 W could be chosen, so as to achieve the desired consensus (KCC, GCC, KGC, GC). The consensus control protocol for the n agents is given by where W ij ∈ R m×m is the weight matrix and [W ij ] kl = w ij kl for k = 1, . . . , m, and l = 126 1, . . . , m. This control law can be written in a compact matrix form as where 128

Remark 3.
For the case when m = 1, the consensus control law becomes: and it can be written in a compact matrix form as:

Remark 4.
For the case when m > 1 and the consensus control law becomes: and it can be written in a compact matrix form as: In order to observe the evolution of x is the kth entry of the control input of agent v i defined in (9). The coefficient w as t → ∞.

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In this case, the evolution of all states follows where u(t) is given in (10). By solving the above differential equation, we have The choice of the weight matrix W will affect the rate of convergence of the con-  The absence of off-diagonal elements in the matrix-weights W ij guarantees that there is no cross-entry state dependence, that is, w ij kl = 0 when k = l. The control law reduces to: u i (t)).
Here we discuss two different cases based on positive definite or positive semi-definite 164 of matrices W ij .
as the vector containing the k-state of all agents. Its dynamics follow The solution of the above differential equation is given by The choice of w ij kk will affect the rate of convergence of the consensus algorithm of the 171 k-state of all agents.

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Consider the case when the matrix weights are scalars multiplying of the identity matrix. We know that for any square matrices A n ∈ R n×n with λ 1 , ..., λ n , and B m ∈ R m×m with µ 1 , ..., µ m that the eigenvalues of A n ⊗ B m are λ i µ j , i = 1, . . . , n, j = 1, . . . , m [18]. Hence, the eigenvalues of L W = L w (G) ⊗ I m are the eigenvalues of L w (G) repeated m times, and L     states between agent v i and agent v j , which leads to C k = C l in general.

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Theorem 4. Assume that the matrix weights are diagonal PSD matrices, then for any state k, 185 the following holds. 186 1. If the k-state graph is connected, there will be k-global consensus (KGC).     For the graph topology shown in Figure 1, the state trajectory is presented in Figure   199 2. It can be seen from Figure 1 that the 2-state graph is connected, and is not dependent on 200 any other state. Therefore, there is a 2-global consensus (2GC):

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The same reasoning applies to the 3-state graph. The graph is connected and 217 depends on agent states from different clusters in C 2 . There will be at least 2 clusters 218 since the control algorithm u 2 satisfies (12) and is connected to only v [3] 3 , there are two clusters in C 3 (Theorem 2).  W 12 = 3I 3×3 , W 32 = 1.9I 3×3 , W 13 = 5I 3×3 .

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In this paper, the role of each element in the matrix-weights of a multi-agent system 248 have been studied under two classifications: diagonal and non-diagonal elements. The following abbreviations are used in this manuscript: