Data-Driven Approximation of Transfer Operators: Naturally Structured Dynamic Mode Decomposition
Data-Driven Approximation of Transfer Operators: Naturally Structured Dynamic Mode Decomposition
复制标题
传递算子的数据驱动逼近：自然结构化动态模式分解
DOI：
10.23919/acc.2018.8431409
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
U. Vaidya
中科院分区：
文献类型：
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作者：
Bowen Huang;U. Vaidya
在本文中，我们提供了一种新算法，用于时间序列数据的线性传递 Koopman 和 Perron-Frobenius 算子的有限维近似。我们认为，这些传递算子的有限维近似的现有方法，例如动态模式分解（DMD）和扩展动态模式分解（EDMD），没有捕获这些算子的两个重要属性，即正性和马尔可夫属性。我们在本文中提出的算法保留了这两个属性。我们将所提出的算法称为自然结构化 DMD，因为它保留了这些算子的固有属性。自然结构化的 DMD 算法可以更好地逼近系统在计算 Koopman 和 Perron-Frobenius 算子本征函数和本征值方面的稳态动力学。然而，保持正性对于捕捉系统的真实瞬态动态至关重要。传递算子的这种正性质及其有限维近似对于非线性系统的控制器和估计器设计起着重要作用。
In this paper, we provide a new algorithm for the finite dimensional approximation of the linear transfer Koopman and Perron-Frobenius operator from time series data. We argue that existing approach for the finite dimensional approximation of these transfer operators such as Dynamic Mode Decomposition (DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two important properties of these operators, namely positivity and Markov property. The algorithm we propose in this paper preserve these two properties. We call the proposed algorithm as naturally structured DMD since it retains the inherent properties of these operators. Naturally structured DMD algorithm leads to a better approximation of the steady-state dynamics of the system regarding computing Koopman and Perron- Frobenius operator eigenfunctions and eigenvalues. However, preserving positivity property is critical for capturing the real transient dynamics of the system. This positivity property of the transfer operators and it's finite dimensional approximation play an important role for controller and estimator design of nonlinear systems.