Balanced Singular Perturbation Approximation (SPA) is a model order reduction method for linear time-invariant systems that guarantees asymptotic stability and for which there exists an a priori error bound. In that respect, it is similar to Balanced Truncation (BT). However, the reduced models obtained by SPA generally introduce better approximation in the lower frequency range and near steady-states, whereas BT is better suited for the higher frequency range. Even so, independently of the frequency range of interest, BT and its variants are more often applied in practice, since there exist more efficient algorithmic realizations thereof. In this paper, we aim at closing this practically-relevant gap for SPA. We propose two novel and efficient algorithms that are adapted for different settings. Firstly, we derive a low-rank implementation of SPA that is applicable in the large-scale setting. Secondly, a data-driven reinterpretation of the method is proposed that only requires input-output data, and thus, is realization-free. A main tool for our derivations is the reciprocal transformation, which induces a distinct view on implementing the method. While the reciprocal transformation and the characterization of SPA is not new, its significance for the practical algorithmic realization has been overlooked in the literature. Our proposed algorithms have well-established counterparts for BT, and as such, also a comparable computational complexity. The numerical performance of the two novel implementations is tested for several numerical benchmarks, and comparisons to their counterparts for BT as well as the existing implementations of SPA are made.

翻译：平衡奇异摄动逼近（SPA）是一种针对线性时不变系统的模型降阶方法，既能保证渐近稳定性，又具有先验误差界。在这方面，它与平衡截断（BT）类似。然而，SPA获取的降阶模型通常在低频段及稳态附近引入更优的逼近效果，而BT则更适用于高频段。尽管如此，无论关注何种频段，BT及其变体在实际中仍被更广泛应用，因其存在更高效的算法实现。本文旨在弥合SPA在这一实践关键领域的差距。我们提出两种适用于不同场景的新型高效算法。首先，推导出适用于大规模场景的SPA低秩实现；其次，提出该方法的数值驱动重新诠释——仅需输入输出数据，从而无需系统实现。推导的核心工具是对偶变换，它为方法实现提供了独特视角。虽然对偶变换与SPA的数学表征并非新概念，但其对实际算法实现的重要性在文献中一直被忽视。所提算法在BT方法中具有成熟的对应版本，因此计算复杂度相当。通过多个数值基准测试验证两种新实现的数值性能，并与BT对应算法及SPA现有实现进行比较。