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.

翻译：将库普曼算子推广至含控制输入的系统并推导非线性基本引理，是发展非线性系统数据驱动控制方法中两个尚未解决的关键问题。本文基于状态与输入可观测量函数希尔伯特空间的张量积，构建乘积希尔伯特空间中的基函数展开，提出了上述问题的新颖解决方案。我们识别出松弛不变性条件，该条件保证从所构建的乘积希尔伯特空间到对应前向传播提升状态希尔伯特空间之间存在有界线性算子（即广义库普曼算子）。与经典库普曼不变性条件相比，本方法无需测度保持性质。此外，通过利用所构建的精确无穷维双线性库普曼表示与汉克尔算子，我们推导出非线性基本引理。所发展的广义库普曼嵌入方法在范德波尔振荡器与软体机械臂模型预测控制中的有效性得到了验证。