Excitation Conditions for Uniform Exponential Stability of the Cooperative Gradient Algorithm over Weakly Connected Digraphs

In this letter, we study the problem of robust adaptive parameter estimation over networks with persistently exciting (PE) nodes and cooperative estimation dynamics. For this problem, it is well known that for networks characterized by undirected connected graphs, the property of uniform exponential stability (UES) can be established under a cooperative PE condition that relaxes the standard individual PE assumptions traditionally used in adaptive control. However, it is an open question whether similar cooperative PE conditions can also be used in general directed graphs. We provide an answer to this question by characterizing a generalized cooperative PE condition that is proved to be necessary and sufficient for UES in cooperative gradient dynamics evolving over arbitrary weakly connected digraphs. We also derive a similar generalized cooperative data-based condition for distributed learning dynamics that use recorded data instead of persistently exciting signals. We further present numerical experiments that study the rates of convergence of the dynamics.