Metrics for ergodicity and design of ergodic dynamics for multi-agent systems

In this paper we propose a metric that quantifies how far trajectories are from being ergodic with respect to a given probability measure. This metric is based on comparing the fraction of time spent by the trajectories in spherical sets to the measure of the spherical sets. This metric is shown to be equivalent to a metric obtained as a distance between a certain delta-like distribution on the trajectories and the desired probability distribution. Using this metric, we formulate centralized feedback control laws for multi-agent systems so that agents trajectories sample a given probability distribution as uniformly as possible. The feedback controls we derive are essentially model predictive controls in the limit as the receding horizon goes to zero and the agents move with constant speed or constant forcing (in the case of second-order dynamics). We numerically analyze the closed-loop dynamics of the multi-agents systems in various scenarios. The algorithm presented in this paper for the design of ergodic dynamics will be referred to as Spectral Multiscale Coverage (SMC).