Robust Weighted-Average Continuous-Time Consensus With Communication Time Delay

Achieving consensus behavior robust to time delay in multiagent systems has attracted much attention. This work is concerned with optimizing the convergence rate of the consensus algorithm in such systems with time delays. Previous approaches optimize either the robustness to time delay or the convergence rate separately, while imposing a limit on the other. Eigenratio optimization is another method, which does not necessarily result in a unique set of weights. Here, the problem is treated in its general form as a multiobjective optimization problem. It is shown that the corresponding Pareto frontier depends solely on the optimal condition number of the Laplacian, and it includes the optimal answer of previously adopted approaches as special cases. A notion of optimal consensusability is then defined, which allows a particular point on the Pareto Frontier with special properties to be identified. The resulting optimization problem is shown to be convex, as is solved by reformulating it as a standard semidefinite programming problem. The optimal weights for individual topologies, clique lifted graphs, and different types of subgraphs are provided, where for the latter, the optimal weights have shown to be independent of the rest of topology. Through numerical simulations, the tradeoff between robustness and convergence rate is demonstrated.