高阶动力系统动力模态分解的理论基础
Theoretical Foundations for the Dynamic Mode Decomposition of High Order Dynamical Systems
摘要：
传统上，高阶动力系统的数据驱动的识别和控制问题是通过用输出的导数来扩充系统状态以在更高维度上用公式表示一阶动力系统来解决的。然而，扩增问题的解决方案通常需要知识的完整的扩增状态，这需要原始输出的数值微分，经常导致噪声信号。这份手稿发展的理论需要直接分析高阶动力系统使用高阶刘维尔算子。这一理论发展的基础是引入信号值RKHS和新的运营商提出了这些空间。最后，据观察，尽管增加了抽象，必要的计算是非常相似的一阶DMD方法使用占用内核。
英文摘要：
Conventionally, data driven identification and control problems for higher order dynamical systems are solved by augmenting the system state by the derivatives of the output to formulate first order dynamical systems in higher dimensions. However, solution of the augmented problem typically requires knowledge of the full augmented state, which requires numerical differentiation of the original output, frequently resulting in noisy signals. This manuscript develops the theory necessary for a direct analysis of higher order dynamical systems using higher order Liouville operators. Fundamental to this theoretical development is the introduction of signal valued RKHSs and new operators posed over these spaces. Ultimately, it is observed that despite the added abstractions, the necessary computations are remarkably similar to that of first order DMD methods using occupation kernels.
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