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 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 1) a nonlinear system with a known linear embedding, 2) a
variety of Hamiltonian systems, and 3) a high-dimensional double-gyre model for
ocean mixing.

这篇论文描述了使用 Koopman 算子及其特征函数进行非线性系统的控制的方法，并提出了基于数据驱动的降阶方法，证明了验证确定的特征函数的重要性并且说明了其在控制中的作用。