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力系统辨识任务，特别针对高频行为非标准的系统——其传递函数具有非平凡相对阶，或等价地，其作为描述符系统的实现具有非平凡指数。我们开发了新颖的代理建模策略，使先进的有理逼近算法（如AAA算法和矢量拟合）能更好地处理来自此类非平凡相对阶系统的数据。我们的贡献体现在两个方面：一方面，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对其进行镜像反映。代理模型所需阶数的实现是通过对其重心系数施加约束，而非通过对有理形式进行临时修改。另一方面，我们提出了一种阶数辨识方法，可从低频数据中估计系统的未知相对阶。通过识别生成数据的系统阶数，我们能够构建出不仅能在低频段良好匹配数据，同时在高频段具有增强外推能力的代理模型。我们通过一系列数值实验验证了新提出方法的有效性与鲁棒性。