线性动力系统分布式在线控制悔值分析的谱滤波方法
A Spectral Filtering Approach to Regret Analysis of Distributed Online Control for Linear Dynamical Systems
- National Cheng Kung University（成功大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文将在线谱控制框架拓展至分布式场景，通过基于Hankel矩阵前导特征向量的谱控制器与分布式在线梯度下降更新，为线性时不变系统分布式在线控制建立了含稳定裕度、网络特性的次线性悔界。
AI中文摘要：
本文研究存在对抗性扰动和时变凸成本时，由线性时不变（LTI）系统组成的网络上的分布式在线控制问题。网络成本由各局部成本函数之和定义，每个局部函数仅按序披露给对应智能体。每个智能体的目标是仅利用局部观测和邻居通信生成控制序列，与事后最优的集中式线性策略相竞争。我们将近期提出的Online Spectral Control（在线谱控制）框架从集中式场景拓展至分布式场景。具体而言，每个智能体采用的谱控制器通过将历史扰动与Hankel矩阵的前导特征向量卷积得到，同时控制器参数通过基于局部替代成本的分布式在线梯度下降步骤更新。我们基于谱参数化将该问题建模为悔值最小化问题，并在标准假设下，建立了$O(\frac{\text{poly}(\text{log}T)}{γ^3})$的次线性悔界，其中$T$为时间跨度，$γ$表示稳定裕度。所得悔界还体现了其与网络规模和连通性的依赖关系。
英文摘要：
This paper studies the distributed online control problem over a network of linear time-invariant (LTI) systems in the presence of adversarial disturbances and time-varying convex costs. The network cost is characterized by the summation of local cost functions, where each local function is sequentially revealed only to the corresponding agent. The goal of each agent is to generate a control sequence, using only local observations and neighbor communication, that competes with the best {\it centralized} linear policy in hindsight. We extend the recently proposed Online Spectral Control framework from the centralized setting to the distributed setting. In particular, each agent applies a spectral controller obtained by convolving past disturbances with the leading eigenvectors of a Hankel matrix, while the controller parameters are updated through a distributed online gradient descent step over the local surrogate costs. We formulate this problem this problem as a {\it regret} minimization problem based on the spectral parameterization, and under standard assumptions, we establish a sublinear regret bound of $O(\frac{\sqrt{T}\text{poly}(\log T)}{γ^3})$, where $T$ is the time horizon and $γ$ denotes the stability margin. The resulting bound also captures the dependence on the network size and connectivity.
