Regret Analysis for Risk-aware Linear Quadratic Control

This paper investigates the regret associated with the Distributionally Robust Control (DRC) strategies used to address multistage optimization problems where the involved probability distributions are not known exactly, but rather are assumed to belong to specified ambiguity families. We quantify the price (distributional regret) that one ends up paying for not knowing the exact probability distribution of the stochastic system uncertainty while aiming to control it using the DRC strategies. The conservatism of the DRC strategies for being robust to worst-case uncertainty distribution in the considered ambiguity set comes at the price of lack of knowledge about the true distribution in the set. We use the worst case Conditional Value-at-Risk to define the distributional regret and the regret bound was found to be increasing with tighter risk level. The motive of this paper is to promote the design of new control algorithms aiming to minimize the distributional regret.