To address the issue of inaccurate distributions in practical stochastic systems, a minimax linear-quadratic control method is proposed using the Wasserstein metric. Our method aims to construct a control policy that is robust against errors in an empirical distribution of underlying uncertainty, by adopting an adversary that selects the worst-case distribution. The opponent receives a Wasserstein penalty proportional to the amount of deviation from the empirical distribution. A closed-form expression of the finite-horizon optimal policy pair is derived using a Riccati equation. The result is then extended to the infinite-horizon average cost setting by identifying conditions under which the Riccati recursion converges to the unique positive semi-definite solution to an algebraic Riccati equation. Our method is shown to possess several salient features including closed-loop stability, and an out-of-sample performance guarantee. We also discuss how to optimize the penalty parameter for enhancing the distributional robustness of our control policy. Last but not least, a theoretical connection to the classical $H_\infty$-method is identified from the perspective of distributional robustness.

中文翻译：

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具有Wasserstein距离的极大极小线性二次控制中的分布鲁棒性。

为了解决实际随机系统中分布不准确的问题，提出了一种使用Wasserstein度量的极大极小线性二次控制方法。我们的方法旨在通过采用选择最坏情况分布的对手来构建对潜在不确定性的经验分布中的错误具有鲁棒性的控制策略。对手会收到与经验分布的偏差量成正比的Wasserstein罚分。使用Riccati方程可导出有限水平最优策略对的闭式表达式。然后，通过确定Riccati递归收敛到代数Riccati方程的唯一正半定解的条件，将结果扩展到无限水平平均成本设置。我们的方法显示出具有多个显着特征，包括闭环稳定性和样本外性能保证。我们还将讨论如何优化惩罚参数以增强控制策略的分布鲁棒性。最后但并非最不重要的一点是，从分布稳健性的角度确定了与经典$ H_ \ infty $方法的理论联系。