Balancing Agility and Communication: Denser Networks Require Faster Agents

This article delves into the challenges of ensuring stability (in some sense) and robustness in large-scale second-order consensus networks (SOCNs) and autonomous vehicle platoons in the discrete-time domain. We propose a graph-theoretic methodology for designing a state feedback law for these systems in a discrete-time framework. By analyzing the behavior of the solutions of the networks based on the algebraic properties of the Laplacian matrices of the underlying graphs and on the value of the update cycle (also known as the time step) of each vehicle, we provide a necessary and sufficient condition for the stability of a linear second-order consensus network in the discrete-time domain. We then perform an $\mathcal {H}_{2}$ -based robustness analysis to demonstrate the relationship between the $\mathcal {H}_{2}$ -norm of the system, network size, connectivity, and update cycles, providing insights into how these factors impact the convergence and robustness of the system. A key contribution of this work is the development of a formal framework for understanding the link between an $\mathcal {H}_{2}$ -based performance measure and the restrictions on the update cycle of the vehicles. Specifically, we show that denser networks (i.e., networks with more communication links) might require faster agents (i.e., smaller update cycles) to outperform or achieve the same level of robustness as sparse networks (i.e., networks with fewer communication links) - see Fig. 1. These findings have important implications for the design and implementation of large-scale consensus networks and autonomous vehicle platoons, highlighting the need for a balance between network density and update cycle speed for optimal performance. We finish the article with results from simulations and experiments that illustrate the effectiveness of the proposed framework in predicting the behavior of vehicle platoons, even for more complex agents with nonlinear dynamics, using Quanser's Qlabs and Qcars.