Data-driven models for nonlinear dynamical systems based on approximating the underlying Koopman operator or generator have proven to be successful tools for forecasting, feature learning, state estimation, and control. It has become well known that the Koopman generators for control-affine systems also have affine dependence on the input, leading to convenient finite-dimensional bilinear approximations of the dynamics. Yet there are still two main obstacles that limit the scope of current approaches for approximating the Koopman generators of systems with actuation. First, the performance of existing methods depends heavily on the choice of basis functions over which the Koopman generator is to be approximated; and there is currently no universal way to choose them for systems that are not measure preserving. Secondly, if we do not observe the full state, we may not gain access to a sufficiently rich collection of such functions to describe the dynamics. This is because the commonly used method of forming time-delayed observables fails when there is actuation. To remedy these issues, we write the dynamics of observables governed by the Koopman generator as a bilinear hidden Markov model, and determine the model parameters using the expectation-maximization (EM) algorithm. The E-step involves a standard Kalman filter and smoother, while the M-step resembles control-affine dynamic mode decomposition for the generator. We demonstrate the performance of this method on three examples, including recovery of a finite-dimensional Koopman-invariant subspace for an actuated system with a slow manifold; estimation of Koopman eigenfunctions for the unforced Duffing equation; and model-predictive control of a fluidic pinball system based only on noisy observations of lift and drag.

翻译：基于近似底层Koopman算子或生成器的非线性动力学系统数据驱动模型，已在预测、特征学习、状态估计与控制领域展现出成功应用。众所周知，控制仿射系统的Koopman生成器对输入具有仿射依赖性，从而可便捷地建立动力学系统的有限维双线性近似。然而，当前近似含驱动系统Koopman生成器的方法仍面临两大障碍。首先，现有方法的性能严重依赖于近似Koopman生成器所采用的基函数选择，且对于非保测系统尚无普适性选择方案。其次，若未能观测到完整状态，则可能无法获取足够丰富的此类函数来描述动力学行为——这是因为当存在驱动时，常用的时间延迟观测量构建方法将失效。为解决上述问题，我们将Koopman生成器支配的观测量动力学建模为双线性隐马尔可夫模型，并采用期望最大化算法确定模型参数。E步涉及标准卡尔曼滤波器与平滑器，M步则类似于生成器的控制仿射动态模态分解。我们通过三个算例验证该方法性能：对具有慢流形的驱动系统恢复有限维Koopman不变子空间；估算无驱杜芬方程的Koopman本征函数；以及仅基于升力与阻力的噪声观测值对流体弹球系统进行模型预测控制。