Analysis and Control of Multi-Time-Scale Modular Directed Heterogeneous Networks

Abstract

In this article, we study the collective behavior of heterogeneous nonlinear systems, interconnected over generic directed graphs and in the scenario that, due to the nature of their interconnections, the agents self-organize in modules. These are subnetworks composed of agents that are densely connected with a strong coupling, while the modules themselves are sparsely interconnected. As we show, beyond certain coupling thresholds, the systems within each module synchronize rapidly with a weighted-average dynamical system that evolves more slowly than the individual systems. Then, the average dynamical systems corresponding to each and all the modules synchronize among themselves. Furthermore, we establish global asymptotic stability for the overall network under the conditions that the average dynamics admit the origin as a globally asymptotically stable equilibrium and each system be semipassive. Finally, we explore stabilization techniques that consist in controlling the average dynamics to make the origin globally asymptotically stable.