高保真非线性结构参数动力系统的优势子空间与模型降阶
Dominant Subspaces of High-Fidelity Nonlinear Structured Parametric Dynamical Systems and Model Reduction
摘要：
在这项工作中，我们研究了高保真非线性结构参数动力系统的模型降阶方案。更具体地说，我们考虑一类非线性动力系统，其非线性项是多项式函数，线性部分对应于线性结构模型，如二阶，时滞，或分数阶系统。我们的方法依赖于这些动力系统的Volterra级数表示。使用这种表示，我们确定的内核，因此，广义多元传递函数与这些系统。因此，我们提出的结果允许建设的降阶模型，其广义传递函数内插这些原系统在预定义的频率点。为了有效的计算，我们还需要张量的对称克罗内克积表示的概念，并推导出它们的特殊性质。此外，我们提出了一个算法，提取占主导地位的子空间，从规定的插值条件。这允许构造保持结构的降阶模型。我们还将这些结果推广到参数系统和一种特殊情况（输入/输出延迟）。我们证明了所提出的方法的效率，通过各种数值基准。
英文摘要：
In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.
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