An Online Learning Analysis of Minimax Adaptive Control

We present an online learning analysis of minimax adaptive control for the case where the uncertainty includes a finite set of linear dynamical systems. Precisely, for each system inside the uncertainty set, we define the model-based regret by comparing the state and input trajectories from the minimax adaptive controller against that of an optimal controller in hindsight that knows the true dynamics. We then define the total regret as the worst case model-based regret with respect to all models in the considered uncertainty set. We study how the total regret accumulates over time and its effect on the adaptation mechanism employed by the controller. Moreover, we investigate the effect of the disturbance on the growth of the regret over time and draw connections between robustness of the controller and the associated regret rate.