On Analytical Construction of Observable Functions in Extended Dynamic Mode Decomposition for Nonlinear Estimation and Prediction
On Analytical Construction of Observable Functions in Extended Dynamic Mode Decomposition for Nonlinear Estimation and Prediction
复制标题
非线性估计和预测的扩展动态模态分解中可观函数的解析构造
作者：
M. Netto;Y. Susuki;V. Krishnan;Y. Zhang
我们提出了一个可观测函数的扩展动态模式分解（EDMD）算法的分析结构。EDMD是一种近似Koopman算子谱特性的数值方法。可观测函数的选择是EDMD应用于系统和控制中的非线性问题的基础。现有的方法要么从一组字典函数开始，寻找最适合底层非线性动力学的子集，要么依赖于机器学习算法来“学习”可观察函数。相反，在这封信中，我们从动力系统模型开始，通过李导数将其提升，将其呈现为多项式形式。这个建议的转换成多项式形式是准确的，它提供了一个足够的一组可观察的功能。所提出的方法的优势是它的适用性更广泛的一类非线性动力系统，特别是那些非多项式函数及其组合。此外，它保留了底层动力系统的物理可解释性，可以很容易地集成到现有的数值库。所提出的方法与电力系统的应用程序进行说明。建模系统由一个单一的发电机连接到一个无限的总线，其中非线性项包括正弦和余弦函数。结果表明，所提出的方法在吸引子外非线性动力学估计和预测的有效性;从所提出的构造获得的可观测函数优于使用包括单项式或径向基函数的字典函数的方法。
We propose an analytical construction of observable functions in the extended dynamic mode decomposition (EDMD) algorithm. EDMD is a numerical method for approximating the spectral properties of the Koopman operator. The choice of observable functions is fundamental for the application of EDMD to nonlinear problems arising in systems and control. Existing methods either start from a set of dictionary functions and look for the subset that best fits the underlying nonlinear dynamics or they rely on machine learning algorithms to “learn” observable functions. Conversely, in this letter, we start from the dynamical system model and lift it through the Lie derivatives, rendering it into a polynomial form. This proposed transformation into a polynomial form is exact, and it provides an adequate set of observable functions. The strength of the proposed approach is its applicability to a broader class of nonlinear dynamical systems, particularly those with nonpolynomial functions and compositions thereof. Moreover, it retains the physical interpretability of the underlying dynamical system and can be readily integrated into existing numerical libraries. The proposed approach is illustrated with an application to electric power systems. The modeled system consists of a single generator connected to an infinite bus, where nonlinear terms include sine and cosine functions. The results demonstrate the effectiveness of the proposed procedure in off-attractor nonlinear dynamics for estimation and prediction; the observable functions obtained from the proposed construction outperform methods that use dictionary functions comprising monomials or radial basis functions.