Data-driven transformations that reformulate nonlinear systems in a linear framework have the potential to enable the prediction, estimation, and control of strongly nonlinear dynamics using linear systems theory. The Koopman operator has emerged as a principled linear embedding of nonlinear dynamics, and its eigenfunctions establish intrinsic coordinates along which the dynamics behave linearly. Previous studies have used finite-dimensional approximations of the Koopman operator for model-predictive control approaches. In this work, we illustrate a fundamental closure issue of this approach and argue that it is beneficial to first validate eigenfunctions and then construct reduced-order models in these validated eigenfunctions. These coordinates form a Koopman-invariant subspace by design and, thus, have improved predictive power. We show then how the control can be formulated directly in these intrinsic coordinates and discuss potential benefits and caveats of this perspective. The resulting control architecture is termed Koopman Reduced Order Nonlinear Identification and Control (KRONIC). It is further demonstrated that these eigenfunctions can be approximated with data-driven regression and power series expansions, based on the partial differential equation governing the infinitesimal generator of the Koopman operator. Validating discovered eigenfunctions is crucial and we show that lightly damped eigenfunctions may be faithfully extracted from EDMD or an implicit formulation. These lightly damped eigenfunctions are particularly relevant for control, as they correspond to nearly conserved quantities that are associated with persistent dynamics, such as the Hamiltonian. KRONIC is then demonstrated on a number of relevant examples, including (a) a nonlinear system with a known linear embedding, (b) a variety of Hamiltonian systems, and (c) a high-dimensional double-gyre model for ocean mixing.

在线性框架中重新构建非线性系统的数据驱动转换有可能使用线性系统理论来预测、估计和控制强非线性动力学。Koopman 算子已成为非线性动力学的原则性线性嵌入，其特征函数建立了动力学线性行为的内在坐标。先前的研究已将 Koopman 算子的有限维近似用于模型预测控制方法。在这项工作中，我们说明了这种方法的一个基本闭合问题，并认为首先验证特征函数然后在这些验证的特征函数中构建降阶模型是有益的。这些坐标通过设计形成了一个库普曼不变子空间从而提高了预测能力。然后我们展示了如何在这些内在坐标中直接制定控制，并讨论这种观点的潜在好处和注意事项。由此产生的控制架构称为Koopman 降阶非线性识别和控制（克罗尼克）。进一步证明，基于控制 Koopman 算子的无穷小生成器的偏微分方程，可以使用数据驱动的回归和幂级数展开来近似这些特征函数。验证发现的特征函数至关重要，我们表明可以从 EDMD 或隐式公式中忠实地提取轻阻尼特征函数。这些轻微阻尼的本征函数与控制特别相关，因为它们对应于与持久动力学相关的近乎守恒的量，例如哈密顿量。然后在许多相关示例中演示了 KRONIC，包括 (a) 具有已知线性嵌入的非线性系统，(b) 各种哈密顿系统，以及 (c) 用于海洋混合的高维双环流模型。