解构控制系统的线性算子学习
Demystifying Linear Operator Learning for Control Systems
- Technical University of Munich（慕尼黑工业大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出利用（半）群和逆问题框架，从数据中结构化学习控制系统线性算子，并通过误差分解和收敛保证推导出可证明优势的算法，为时变系统提供收敛估计器。
AI中文摘要：
本文提出了一种从数据中学习控制系统线性算子的结构化方法。我们同时处理该问题的结构性和学习理论性方面。为了推导结构性假设，我们建议使用成熟的演化方程（半）群框架，因为控制系统中的算子属于同一类型。此外，我们建议通过逆问题框架的视角来分析学习算法。这通过误差分解、收敛保证和最优正则化揭示了学习模型如何依赖于数据——使我们能够比较现有方法并推导出可证明具有优势的算法。为了获得这些结果，我们将范围限制在希尔伯特空间上的有界算子。尽管这看起来具有限制性，但现有方法通常隐式地做出这一假设以获得类似矩阵的表示。我们通过为时变系统推导一个收敛估计器来展示使用这些框架的威力。
英文摘要：
This paper proposes a structured approach to learning linear operators for control systems from data. We address both structural and learning-theoretic aspects of the problem. To derive structural assumptions, we propose using the well-established framework of (semi)groups for evolution equations, as operators in control systems are of the same type. Further, we propose analyzing learning algorithms through the lens of the inverse problems framework. This reveals how a learned model depends on the data via error decompositions, convergence guarantees, and optimal regularization -- enabling us to compare existing methods and derive provably advantageous algorithms. In order to obtain these results, we restrict our scope to bounded operators on Hilbert spaces. Although this may appear restrictive, existing approaches often make this assumption implicitly to obtain matrix-like representations. We demonstrate the power of using these frameworks by deriving a convergent estimator for time-varying systems.
