Distributed PID Control for Consensus and Synchronization of Multi-agent Networks

We investigate the use of distributed PID actions to achieve consensus and synchronization in networks of homogeneous and heterogeneous agents. We first analyze the case of distributed PID control on networks with heterogeneous nodes described by first-order linear systems. Convergence of the strategy is proved using appropriate state transformations and Lyapunov functions. Then, we propose a multiplex proportional-integral approach, for solving consensus problems in networks of heterogeneous n-dimensional node dynamics affected by constant disturbances. The proportional and integral actions are deployed on two different layers across the network, each with its own topology. Furthermore, the contribution of the network topology and node dynamics have been systematically separated giving some sufficient conditions guaranteeing convergence. Finally, an extension to networks of identical nonlinear node dynamics is presented. We provide local and global stability analysis together with a detailed performance assessment where heterogeneity among nodes and disturbances are considered. The effectiveness of the theoretical results is illustrated via its application to a representative power grid model recently presented in the literature and also for synchronization in networks of chaotic circuits.