We consider the task of data-driven identification of dynamical systems, specifically for systems whose behavior at large frequencies is non-standard, as encoded by a non-trivial relative degree of the transfer function or, alternatively, a non-trivial index of a corresponding realization as a descriptor system. We develop novel surrogate modeling strategies that allow state-of-the-art rational approximation algorithms (e.g., AAA and vector fitting) to better handle data coming from such systems with non-trivial relative degree. Our contribution is twofold. On one hand, we describe a strategy to build rational surrogate models with prescribed relative degree, with the objective of mirroring the high-frequency behavior of the high-fidelity problem, when known. The surrogate model's desired degree is achieved through constraints on its barycentric coefficients, rather than through ad-hoc modifications of the rational form. On the other hand, we present a degree-identification routine that allows one to estimate the unknown relative degree of a system from low-frequency data. By identifying the degree of the system that generated the data, we can build a surrogate model that, in addition to matching the data well (at low frequencies), has enhanced extrapolation capabilities (at high frequencies). We showcase the effectiveness and robustness of the newly proposed method through a suite of numerical tests.

翻译：本文研究数据驱动的动态系统辨识任务，特别针对高频行为非标准的系统——此类特性体现为传递函数具有非平凡相对阶，或等价地表现为描述符系统实现具有非平凡指数。我们开发了新颖的代理建模策略，使前沿有理逼近算法（如AAA算法和向量拟合）能更有效地处理来自此类非平凡相对阶系统的数据。我们的贡献包含两个方面：首先，我们提出了一种构建具有指定相对阶的有理代理模型的策略，旨在当已知高保真问题的高频行为时实现对其的准确映射。该策略通过约束重心系数而非临时修改有理形式来实现代理模型的期望阶数。其次，我们提出了一种阶数辨识方法，能够根据低频数据估计系统的未知相对阶。通过识别生成数据的系统阶数，我们构建的代理模型不仅能在低频段良好匹配数据，还能在高频段具备增强的外推能力。我们通过一系列数值实验验证了新提出方法的有效性和鲁棒性。