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, then it becomes necessary to account for the dependence of the output time series on the sequence of supplied inputs when constructing observables to approximate Koopman operators. To address 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：基于逼近底层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、状态估计和控制领域展现出成功应用。众所周知，控制仿射系统的Koopman生成器也依赖于输入的仿射关系，从而可对动力学进行便捷的有限维双线性近似。然而，当前逼近驱动系统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用于逼近Koopman生成器的基函数选择；对于非保测系统，目前尚无通用方法可进行选择。其次，若未观测到完整状态，则在构建可观测量以近似Koopman算子时，需考虑输出时间序列对输入序列的依赖性。为解决这些问题，我们将Koopman生成器控制的可观测量动力学建模为双线性隐马尔可夫模型，并使用期望最大化（EM）算法确定模型参数。其中，E步采用标准卡尔曼滤波和平滑算法，M步则类似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：包括含慢流形驱动系统的有限维Koopman不变子空间恢复、无强迫Duffing方程的Koopman本征函数估计，以及仅基于升力和阻力含噪观测的流管球系统模型预测控制。