The extended dynamic mode decomposition algorithm is a tool for accurately approximating the point spectrum of the Koopman operator. This algorithm provides an approximate linear expansion of non-linear discrete-time systems, which can be useful for system analysis and controller design. The accuracy of this algorithm depends heavily on the availability of a set of basis functions that provide the ability to capture the nonlinear dynamics of the underlying system. Recently, the use of orthogonal polynomials, along with reduction techniques for the dimension and maximum order of the polynomial basis, have been successfully used to approximate nonlinear systems with the additional benefit of using smaller datasets. This paper expands the current methods for selecting the set of observables for nonlinear systems with periodic behavior, which is prone to a representation in terms of trigonometric functions. The benefit of working with orthogonal polynomials is preserved by embedding the trigonometric functions into the orthogonal basis. The algorithm is illustrated with the data-driven modelling of an inverted pendulum in simulation and real-life experiments.

中文翻译：

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多项式扩展模式分解中的三角嵌入—在倒立摆上的实验应用

扩展动态模式分解算法是一种用于精确逼近Koopman算子的点谱的工具。该算法提供了非线性离散时间系统的近似线性展开，这对于系统分析和控制器设计很有用。该算法的准确性在很大程度上取决于一组基础函数的可用性，这些基础函数提供了捕获底层系统的非线性动力学的能力。最近，正交多项式的使用以及多项式基础的维数和最大阶数的归约技术已被成功地用于近似非线性系统，并具有使用较小数据集的额外好处。本文扩展了选择具有周期性行为的非线性系统的可观测量集的当前方法，这很容易用三角函数来表示。通过将三角函数嵌入正交基，可以保留使用正交多项式的好处。在仿真和实际实验中，以倒立摆的数据驱动建模来说明该算法。