$\mathcal{H}_2$ 综合的良性几何与分布鲁棒性
Benign Geometry and Distributional Robustness of $\mathcal{H}_2$ Synthesis
- Alpha Brain Technologies（Alpha Brain 技术）
- Delft University of Technology（代尔夫特理工大学）
- University of Toronto（多伦多大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文研究离散时间线性时不变系统的标准与分布鲁棒 $\mathcal{H}_2$ 综合，证明平稳性等价于全局最优性，并推广至分布鲁棒情形，最后刻画三类模糊集的最坏情况协方差以提供性能保证。
中文摘要 AI 辅助
本文研究了离散时间线性时不变系统中镇定状态反馈控制器的标准与分布鲁棒 $\mathcal{H}_2$ 综合问题。在不要求非奇异扰动可控性 Gramian 或正定控制惩罚的情况下，我们建立了标准 $\mathcal{H}_2$ 综合中镇定增益的平稳性等价于全局最优性，并推导了性能差异的精确勾股恒等式。随后，我们通过考虑具有零均值和一致有界二阶矩的独立同分布扰动来推广 $\mathcal{H}_2$ 综合问题，并将平稳性-最优性等价性扩展到相应的分布鲁棒 $\mathcal{H}_2$ 综合。此外，我们证明了每个标准 $\mathcal{H}_2$ 最优增益同时对于每个此类分布鲁棒问题都是最优的。最后，我们将该框架专门应用于 Frobenius、Kullback-Leibler 和 Wasserstein-2 模糊集，并获得了它们最坏情况协方差的显式刻画，这反过来为常见的最优控制器提供了性能保证。
英文摘要
In this paper, we study standard and distributionally robust $\mathcal{H}_2$ synthesis problem of a stabilizing state-feedback controller for discrete-time linear time-invariant systems. Without requiring a nonsingular disturbance controllability Gramian or a positive-definite control penalty, we establish that stationarity of a stabilizing gain in standard $\mathcal{H}_2$ synthesis is equivalent to global optimality and derive an exact Pythagorean identity for the performance difference. We then generalize the $\mathcal{H}_2$ synthesis problem by considering i.i.d. disturbances with zero mean and uniformly bounded second moments, and extend the stationarity-optimality equivalence to the corresponding distributionally robust $\mathcal{H}_2$ synthesis. Moreover, we show that every standard $\mathcal{H}_2$-optimal gain is simultaneously optimal for every such distributionally robust problem. Finally, we specialize the framework to Frobenius, Kullback--Leibler, and Wasserstein-2 ambiguity sets and obtain explicit characterizations of their worst-case covariances, which, in turn, provide performance certificates for a common optimal controller.
