提出新型非线性系统数据驱动控制框架，突破传统限制。
From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma
- 构建状态与输入函数的张量积希尔伯特空间，推导广义科波曼算子。
- 放宽不变性条件，无需测度保持即可保证算子有界。
- 首次实现非线性系统的精确无限维双线性表示，适用于软体机器人控制。
将科波曼算子推广至含控制输入的系统，并推导非线性基础引理，是发展非线性系统数据驱动控制方法的关键挑战。本文基于状态与输入可观测函数各自希尔伯特空间的张量积构造产品希尔伯特空间，通过基函数展开提出新解法。识别出弱化不变性条件，确保从该产品空间到向前推进的提升状态空间的有界线性算子（即广义科波曼算子）存在，相比经典科波曼不变性条件，不再要求测度保持。此外，利用所构建的精确无限维双线性科波曼表示及汉克尔算子，推导出非线性基础引理。所提出的广义科波曼嵌入在范德波尔振子和软体机械臂模型的预测控制中验证了有效性。
原文摘要 · Abstract (English)
The generalization of the Koopman operator to systems with control input and the derivation of a nonlinear fundamental lemma are two open problems that play a key role in the development of data-driven control methods for nonlinear systems. In this paper we derive a novel solution to these problems based on basis functions expansion in a product Hilbert space constructed as the tensor product between the Hilbert spaces of the state and input observable functions, respectively. We identify relaxed invariance conditions that guarantee existence of a bounded linear operator, i.e., the generalized Koopman operator, from the constructed product Hilbert space to the Hilbert space corresponding to the lifted state propagated forward in time. Compared to classical Koopman invariance conditions, measure preservation is not required. Moreover, we derive a nonlinear fundamental lemma by exploiting the constructed exact infinite-dimensional bilinear Koopman representation and Hankel operators. The effectiveness of the developed generalized Koopman embedding is illustrated on the Van der Pol oscillator and in predictive control of a soft-robotic manipulator model.
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