Large-scale linear, time-invariant (LTI) dynamical systems are widely used to characterize complicated physical phenomena. We propose a two-stage algorithm to reduce the order of a large-scale LTI system given samples of its transfer function for a target degree $k$ of the reduced system. In the first stage, a modified adaptive Antoulas--Anderson (AAA) algorithm is used to construct a degree $d$ rational approximation of the transfer function that corresponds to an intermediate system, which can be numerically stably reduced in the second stage using ideas from the theory on Hankel norm approximation (HNA). We also study the numerical issues of Glover's HNA algorithm and provide a remedy for its numerical instabilities. A carefully computed rational approximation of degree $d$ gives us a numerically stable algorithm for reducing an LTI system, which is more efficient than SVD-based algorithms and more accurate than moment-matching algorithms.

翻译：大规模线性时不变（LTI）动力学系统被广泛用于刻画复杂物理现象。针对给定传递函数采样点的目标降阶系统阶数$k$，我们提出一种两阶段算法来降低大规模LTI系统的阶数。在第一阶段，采用改进的自适应Antoulas-Anderson（AAA）算法构造传递函数的$d$阶有理逼近，该逼近对应一个中间系统；在第二阶段，利用Hankel范数逼近（HNA）理论中的思想对该中间系统进行数值稳定的降阶处理。我们还研究了Glover HNA算法的数值问题，并针对其数值不稳定性提出了补救措施。精心计算的$d$阶有理逼近为我们提供了一种数值稳定的LTI系统降阶算法，该算法比基于奇异值分解（SVD）的算法更高效，且比矩匹配算法更精确。