Adaptive consensus and tracking consensus of multi-agent systems with matrix-weighted networks encompassing cooperative and antagonistic relationships

This paper addresses the consensus and tracking consensus problems for multi-agent systems operating over matrix-weighted networks. The networks with only cooperative relationships and with both cooperative and antagonistic relationships are both considered. Matrix-weighted networks can not only describe the relationships among neighbouring agents but also accurately reveal the high-dimensional logical inter-dependency among the states of each agent itself. Novel adaptive control strategies are integrated to design two kinds of distributed protocols. For the leaderless case, the edge-based adaptive consensus protocols are proposed. For the leader-following case, the node-based adaptive tracking consensus protocols are proposed. The node-based counterpart can significantly reduce parameter complexity by employing a single adaptive gain for each agent. Moreover, a nonlinear robust term is introduced into the node-based protocols to ensure tracking consensus convergence even when the leader's input is only required to be bounded but the bound is unknown. Sufficient conditions for achieving consensus and tracking consensus are established based on Lyapunov stability theory. Specially, in the leader-following case, the positive definiteness of the sum of leader-following connection weight matrices is identified as a core verifiable algebraic condition for tracking consensus convergence. Finally, numerical simulations corroborate the effectiveness of the proposed control strategies.