On the Regret of $\mathcal{H}_{\infty}$ Control

The $\mathcal{H}_{\infty}$ synthesis approach is a cornerstone robust control design technique, but is known to be conservative in some cases. The objective of this paper is to quantify the additional cost the controller incurs planning for the worst-case scenario, by adopting an approach inspired by regret from online learning. We define the \textit{disturbance-reality gap} as the difference between the predicted worst-case disturbance signal and the actual realization. The regret is shown to scale with the norm of this \textit{gap}, which turns out to have a similar structure to that of the certainty equivalent controller with inaccurate predictions, obtained here in terms of the \textit{prediction error} norm.