Optimization of convergence rate and stability margin of information flow in cooperative systems

The interplay between the convergence rate and stability margin (e.g. ability to reject disturbances) for a discrete-time information flow filter in cooperative systems is analyzed. For a given communication graph, the convergence rate is defined as the absolute value of the largest nonunit characteristic root of a matrix associated with the filter. The maximal convergence rate, obtained by “tuning” the control gains, is highly correlated to the number of distinct eigenvalues of the graph Laplacian (it is 1 for the complete graph). A stability margin is introduced for multiple-input–multiple-output (MIMO) systems and is then maximized with respect to the control gains subject to a constraint on the convergence rate. The optimal stability margin as a function of the convergence rate is bounded above for any order of the filter, and the bound is attained for the complete graph. For the zero-order filter and all strongly connected communication graphs, the optimal stability margin is found analytically, whereas for the first-order filter and undirected communication graphs, it is evaluated numerically. The results demonstrate the ability to distinguish graph topologies that dominate others in their ability to reject disturbances and converge rapidly to a consensus.