# On the Role of Matrix-Weights Elements in Consensus Algorithms for Multi-Agent Systems
###### Abstract  
This paper examines the roles of the matrix weight elements in matrix-weighted consensus. The consensus algorithms dictate that all agents reach consensus when the weighted graph is connected. However, it is not always the case for matrix weighted graphs. The conditions leading to different types of consensus have been extensively analysed based on the properties of matrix-weighted Laplacians and graph theoretic methods. However, in practice, there is concern on how to pick matrix-weights to achieve some desired consensus, or how the change of elements in matrix weights affects the consensus algorithm. By selecting the elements in the matrix weights, different clusters may be possible. In this paper, we map the roles of the elements of the matrix weights in the systems consensus algorithm. We explore the choice of matrix weights to achieve different types of consensus and clustering. Our results are demonstrated on a network of three agents where each agent has three states.  
matrix-weighted graphs; multi-agent systems; clustered consensus; global consensus +
Footnote †: 2021: 15 October 2021  
of the graph Laplacian. In [15], discrete-time matrix-weighted consensus is studied over undirected and connected graphs considering symmetric matrix weights and a special case of non-symmetric matrix-weights that can achieve consensus control. Lyapunov stability theory for discrete-time systems is also employed to show the system's convergence to consensus. In the same way, in [14], matrix-weighted consensus algorithm is studied with fixed undirected graphs and a necessary and sufficient condition for exponentially reaching a global average consensus and clustered consensus is presented based on the null-space of the matrix-weighted Laplacian.  
Generally, when the agent-to-agent link is weighted by positive semi-definite (PSD) matrices, it can be difficult to tell what form of consensus will be present. Clustered consensus can happen even when the graph is connected in the agent level, due to the existence of PSD matrix weights. A typical example, where all the elements of each of the matrix-weights are set to one, for a complete graph on five vertices, there will be no form of consensus for the agents in any of the states.  
In this paper, we study how the choice of matrix weights for the inter-agent links affect the presence of clusters in a matrix-weighted graph. A linear algebraic approach is used in [17; 18] to examine the properties of the Laplacian matrix and the necessary and sufficient conditions are given for the existence of consensus. We adopt a matrix-element-mapping approach to study how the matrix-weights alter the consensus dynamics. Our approach compares the matrix-based approach to the conventional scalar weighted graphs which helps view the network of each state, and offers hints on the choice of the matrix weights.  
The main contributions of this paper are summarized as follows:  
* to examine the matrix-weighted graph and show how each element influences the consensus algorithm;
* to study how different forms of consensus can be achieved by the choice of matrix weights;
* to be able to determine the number of clusters and the elements in each of the cluster by looking at the matrix-weight set.