标题： 随机线性系统之间的距离 显示英文标题

标题： Distance Between Stochastic Linear Systems

摘要： 本文提出了一种随机线性动态系统之间的距离度量。 尽管现有的随机控制理论已经能够处理具有随机不确定性的动态系统，但为了有效进一步探索该领域，需要采用基于距离度量的决策范式。 作为第一步，这里提出了两个线性时不变随机动态系统之间的距离度量，扩展了现有确定性线性动态系统之间的距离度量。 在频域设置中，提出了随机系统的距离度量，即使用黎曼球面上的反立体投影，表征不确定性分布的最坏情况频率点上的Wasserstein距离。 在时域设置中，所提出的距离对应于在参数空间到各自系统植物空间的可测映射下，两个系统推动前测度之间的类型-$q$Wasserstein距离。 通过数值仿真证明并展示了所提出的频域距离度量永远不会超过所提出的时域距离度量的对应值。 提供了所提出的距离度量在频域和时域设置中的下界和上界。 所提出的距离度量在相应的（频域/时域）随机动态系统空间中诱导了一个拓扑结构，并将有助于提供系统鲁棒性和控制器性能的概率保证。 显示英文摘要

摘要： This manuscript proposes a distance measure between stochastic linear dynamical systems. While the existing stochastic control theory is well equipped to handle dynamical systems with stochastic uncertainties, a paradigm shift using distance measure based decision making is required for the effective further exploration of the field. As a first step, a distance measure between two linear time invariant stochastic dynamical systems is proposed here, extending the existing distance metrics between deterministic linear dynamical systems. Distance measure for stochastic systems is proposed for the frequency domain setting as the worst-case point-wise in frequency Wasserstein distance between distributions characterising the uncertainties using inverse stereographic projection on the Riemann sphere. For the time domain setting, the proposed distance corresponds to the gap metric induced type-$q$ Wasserstein distance between the push-forward measures under both systems' corresponding measurable maps from the parameter space to their respective space of system plants. It is proved and demonstrated using numerical simulation that the proposed frequency domain distance measure shall never exceed the proposed time domain distance measure counterpart. Lower and upper bounds are provided for the proposed distance measures in both frequency and time domain settings. The proposed distance measures induce a topology in the corresponding (frequency/time) domain space of stochastic dynamical systems and will facilitate the provision of probabilistic guarantees on system robustness and controller performances.