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。该算法在大规模线性时不变(LTI)系统的数据驱动模型降阶(MOR)中取得了显著成功，其典型优势在于能够高效且自动化地生成降阶模型(ROM)。尽管经典AAA算法具备有效性与数值可靠性，但无法保证返回的ROM保留原动力系统的结构特征（如动力学稳定性）。本文提出了一种针对传递函数满足AAA重心结构的ROM稳定性的新型代数刻画，并基于该刻画构建了关于AAA模型自由系数的凸约束条件集——当这些条件满足时，可确保所得ROM的渐近稳定性。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算例实验验证了所提算法（命名为stabAAA）的有效性。