Cluster consensus for networks with antagonistic interactions

Consensus refers to the problem of distributed control protocols that enable all agents in a network to reach an agreement on a certain variable or quantity. Cluster consensus is a more generalized form of consensus where, instead of all agents converging to a single common value, they divide themselves into multiple clusters, and agents within each cluster reach an agreement on a common value. This thesis addresses the cluster consensus problem in multi-agent networks operating over arbitrary directed graphs, both unsigned and signed, without imposing restrictive structural assumptions common in existing literature. The research investigates cluster formation and stability across diverse system dynamics, including higher-order linear, time-delayed, inherent nonlinear, and matrix-weighted systems. A major contribution is the development of a systematic framework, notably utilizing an extended graph representation, which enables the explicit determination of the number of clusters and their members for any given signed directed graph. The thesis provides necessary and sufficient conditions, stability analysis, and control design methods for these general networks and system types. Key findings include the explicit determination of clusters for arbitrary signed networks, handling delays and nonlinearities on general graphs, and analyzing matrix-weighted consensus without topological constraints. By overcoming restrictive structural assumptions and providing specific methods for cluster analysis on general directed networks, this research significantly enhances the understanding and solvability of cluster consensus problems for realistic multi-agent system applications. The theoretical results are supported by numerical examples.