Consensus control and optimization analysis of multi-agent system with time-varying delay based on the Lyapunov-Krasovskii functional: Novel order-reduction methods

This paper investigates consensus control and controller-gain optimisation for multi-agent systems with time-varying delays. To mitigate the high-dimensional linear matrix inequalities (LMIs) induced by Kronecker products, two novel order-reduction methods are developed. These methods integrate the controller design and optimisation within a unified framework. These techniques simplify two zero-equation structures structures with free-weighting matrices (including a new type related to Laplacian matrix) and significantly alleviating computational dimension. A Lyapunov-Krasovskii functional is constructed to simultaneously assess and facilitate the controller design, and the proposed reductions markedly decrease conservatism in the consensus conditions. Specifically, one proposed reduction method directly yields a suitable controller without iterative optimization, whereas the other embeds a particle-swarm optimisation (PSO) strategy into the synthesis process. Furthermore, the paper considers both types of zero equations and presents a hybrid approach that combines the advantages of the two order-reduction methods. Even if the number of decision variables increases, the resulting methodology substantially reduces LMI dimensionality. Simulation studies confirm the theoretical effectiveness and applicability of the integrated control and optimization framework.