标题： 从乘积希尔伯特空间到广义Koopman算子和非线性基本引理 显示英文标题

标题： From Product Hilbert Spaces to the Generalized Koopman Operator and the Nonlinear Fundamental Lemma

摘要： 广义Koopman算子在具有控制输入的系统中的推广以及非线性基本引理的推导是两个在发展非线性系统的数据驱动控制方法中起关键作用的开放问题。 这两个问题都依赖于可观测函数或基函数及其对应希尔伯特空间的构造，这些空间能够实现无限维线性系统表示。 在本文中，我们基于状态和输入可观测函数的希尔伯特空间之间的张量积构建的产品希尔伯特空间中的正交展开，推导出这些问题的新解决方案。 我们证明存在一个无限维线性算子，即广义Koopman算子，从构建的产品希尔伯特空间到对应于向前传播提升状态的希尔伯特空间。 还提出了一种可扩展的数据驱动方法，用于计算广义Koopman算子的有限维近似以及几种可观测函数的选择。 此外，我们通过利用无限维广义Koopman模型的双线性结构推导出一个非线性基本引理。 所开发的广义Koopman嵌入的有效性在Van der Pol振荡器上得到了说明。 显示英文摘要

摘要： 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. Both problems hinge on the construction of observable or basis functions and their corresponding Hilbert space that enable an infinite-dimensional, linear system representation. In this paper we derive a novel solution to these problems based on orthonormal expansion in a product Hilbert space constructed as the tensor product between the Hilbert spaces of the state and input observable functions, respectively. We prove that there exists an infinite-dimensional 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. A scalable data-driven method for computing finite-dimensional approximations of generalized Koopman operators and several choices of observable functions are also presented. Moreover, we derive a nonlinear fundamental lemma by exploiting the bilinear structure of the infinite-dimensional generalized Koopman model. The effectiveness of the developed generalized Koopman embedding is illustrated on the Van der Pol oscillator.