Abstract
This paper studies the cluster consensus problem for networks with antagonistic interactions modelled by adjacency matrices with negative weights. By introducing an extended digraph representation consisting of purely cooperative interactions with lifting approach, we relate the trajectories of the system to those of an extended system with a positive digraph. The behaviours of agents in a signed network are extracted from its extended digraph. Consequently, the number of clusters and the cluster members are explicitly determined for any signed digraph by using primary and secondary layer subgraph concepts. The relation between the dynamics of a signed system and its extended representation is investigated for first and higher-order systems. The conditions that make both systems stable are stated. Additionally, the control parameters to achieve cluster consensus are derived explicitly for first, second and third order systems. The obtained results for continuous-time networks are subsequently extended to discrete-time networks. Finally, theoretical results are illustrated via numerical examples.
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