In recent years, the Adaptive Antoulas-Anderson AAA algorithm has established itself as the method of choice for solving rational approximation problems. Data-driven Model Order Reduction (MOR) of large-scale Linear Time-Invariant (LTI) systems represents one of the many applications in which this algorithm has proven to be successful since it typically generates reduced-order models (ROMs) efficiently and in an automated way. Despite its effectiveness and numerical reliability, the classical AAA algorithm is not guaranteed to return a ROM that retains the same structural features of the underlying dynamical system, such as the stability of the dynamics. In this paper, we propose a novel algebraic characterization for the stability of ROMs with transfer function obeying the AAA barycentric structure. We use this characterization to formulate a set of convex constraints on the free coefficients of the AAA model that, whenever verified, guarantee by construction the asymptotic stability of the resulting ROM. We suggest how to embed such constraints within the AAA optimization routine, and we validate experimentally the effectiveness of the resulting algorithm, named stabAAA, over a set of relevant MOR applications.

翻译：近年来，自适应Antoulas-Anderson（AAA）算法已成为求解有理逼近问题的首选方法。面向大规模线性时不变（LTI）系统的数据驱动模型降阶（MOR）是该算法成功应用的众多场景之一——它通常能以自动化方式高效生成降阶模型（ROM）。尽管经典AAA算法在有效性与数值可靠性方面表现优异，但其无法保证生成的ROM保留原始动力系统的结构特征（如动力学稳定性）。本文提出一种针对传递函数服从AAA重心结构的ROM稳定性的新型代数表征，并利用该表征构建关于AAA模型自由系数的凸约束集——当这些约束得到满足时，可确保最终ROM具有渐近稳定性。我们进一步阐述了如何在AAA优化框架中嵌入此类约束，并通过一系列相关MOR应用实验验证了所提算法stabAAA的有效性。