This paper proposes a distributionally robust approach to regret optimal control of discrete-time linear dynamical systems with quadratic costs subject to a stochastic additive disturbance on the state process. The underlying probability distribution of the disturbance process is unknown, but assumed to lie in a given ball of distributions defined in terms of the type-2 Wasserstein distance. In this framework, strictly causal linear disturbance feedback controllers are designed to minimize the worst-case expected regret. The regret incurred by a controller is defined as the difference between the cost it incurs in response to a realization of the disturbance process and the cost incurred by the optimal noncausal controller which has perfect knowledge of the disturbance process realization at the outset. Building on a well-established duality theory for optimal transport problems, we derive a reformulation of the minimax regret optimal control problem as a tractable semidefinite program. Using the equivalent dual reformulation, we characterize a worst-case distribution achieving the worst-case expected regret in relation to the distribution at the center of the Wasserstein ball. We compare the minimax regret optimal control design method with the distributionally robust optimal control approach using an illustrative example and numerical experiments.

翻译：本文针对具有二次成本的离散时间线性动力系统，提出了一种分布鲁棒的遗憾最优控制方法，该系统受状态过程随机加性扰动的影响。扰动过程的潜在概率分布未知，但假设位于由第二类Wasserstein距离定义的给定分布球内。在此框架下，设计严格因果的线性扰动反馈控制器以最小化最坏情况下的期望遗憾。控制器产生的遗憾定义为：控制器对扰动过程实现响应的成本与具有完美知识的最优非因果控制器（该控制器在初始时刻完全了解扰动过程实现）所产生成本之间的差值。基于最优传输问题中成熟的對偶理论，我们将极小极大遗憾最优控制问题重新表述为可处理的半定规划。利用等价的對偶重新表述，我们刻画了实现最坏情况期望遗憾的极值分布与Wasserstein球中心分布之间的关系。通过一个说明性示例和数值实验，我们将极小极大遗憾最优控制设计方法与分布鲁棒最优控制方法进行了比较。