Koopman算子理论：基础、控制与应用

中文摘要 AI 辅助

Koopman算子因其能够提供高度复杂动力系统的全局线性表示而受到广泛关注。该算子通过实值或复值可观测量函数，以线性方式描述非线性动力学。最近提出的数据驱动技术，如扩展动态模式分解（EDMD）、其核化变体以及机器学习方法，可用于生成有限维近似，并附带有限数据误差界。在本教程论文中，我们简要介绍Koopman算子理论及其在系统与控制中的应用。特别关注数据驱动替代模型、其向有输入系统的扩展，以及使用Koopman算子理论的控制器设计。此外，我们演示了关键技术，即EDMD和Koopman MPC。为此，我们提供包括GitHub上源代码的仿真研究，使感兴趣的读者能够逐步体验Koopman算子在系统与控制中的应用。

英文摘要

The Koopman operator has gained considerable attention due to its ability to provide a global linear representation of highly complex dynamical systems. The operator describes nonlinear dynamics in a linear way through the lens of real- or complex-valued observable functions. Data-driven techniques, like extended dynamic mode decomposition (EDMD), kernel EDMD, and machine-learning methods, can be used to generate finite-dimensional approximations accompanied by finite-data error bounds. In this tutorial paper, we provide a concise introduction into Koopman operator theory and its use in systems and control. A particular focus is put on data-driven surrogate models, their extension to systems with inputs, and controller design using Koopman operator theory. Moreover, we demonstrate the key techniques, i.e., EDMD and Koopman MPC. To this end, we provide simulation studies including source code on GitHub to enable the interested reader to experience the Koopman operator in systems and control step by step.

发表机构

• Department of Mechanical Engineering, University of California, Santa Barbara（加州大学圣芭芭拉分校机械工程系）
• Department of Mechanical and Aerospace Engineering, University of California, San Diego（加州大学圣地亚哥分校机械与航空航天工程系）
• Optimization-based Control Group, TU Ilmenau（图林根工业大学优化控制组）
• Constrained Control of Complex Systems Lab, Control Systems Group, Department of Electrical Engineering, TU Eindhoven（埃因霍温理工大学复杂系统约束控制实验室，控制系统组，电气工程系）
• Department of Electrical and Computer Engineering, National University of Singapore（新加坡国立大学电气与计算机工程系）

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