SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 981-996, January 2021.
We propose a computational approach to estimate the stability domain of quadratic-bilinear reduced-order models (ROMs), which are low-dimensional approximations of large-scale dynamical systems. For nonlinear ROMs, it is important not only to show that the origin is locally asymptotically stable, but also to quantify if the operative range of the ROM is included in the region of convergence. While accuracy and structure preservation remain the main focus of development for nonlinear ROMs, computational methods that go beyond the existing highly conservative analytical results have been lacking thus far. In this work, for a given quadratic Lyapunov function, we first derive an analytical estimate of the stability domain, which is rather conservative but can be evaluated efficiently. With the goal to enlarge this estimate, we provide an optimal ellipsoidal estimate of the stability domain by solving a convex optimization problem. This provides us with valuable information about stability properties of the ROM, an important aspect of predictive simulation. We do not assume a specific ROM method, so a particular appeal is that the approach is applicable to quadratic-bilinear models obtained via data-driven approaches, where ROM stability properties cannot---per definition---be derived from the full-order model. Numerical results for a linear quadratic Gaussian (LQG)-balanced ROM of Burgers' equation, a proper orthogonal decomposition ROM of FitzHugh--Nagumo, and a nonintrusive ROM of Burgers' equation demonstrate the scalability and quantitative advantages of the proposed approach. The optimization-based estimates of the stability domain are found to be up to four orders of magnitude less conservative than analytical estimates.

中文翻译：

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二次双线性降阶模型的稳定域

SIAM Journal on Applied Dynamical Systems，第 20 卷，第 2 期，第 981-996 页，2021 年 1 月。
我们提出了一种计算方法来估计二次双线性降阶模型 (ROM) 的稳定域，ROM 是大规模动态系统的低维近似。对于非线性 ROM，重要的是不仅要证明原点是局部渐近稳定的，而且要量化 ROM 的操作范围是否包含在收敛区域中。虽然精度和结构保持仍然是非线性 ROM 开发的主要重点，但迄今为止，缺乏超越现有高度保守分析结果的计算方法。在这项工作中，对于给定的二次李雅普诺夫函数，我们首先推导出稳定域的解析估计，这是相当保守但可以有效评估的。为了扩大这个估计，通过解决凸优化问题，我们提供了稳定域的最佳椭圆估计。这为我们提供了有关 ROM 稳定性特性的宝贵信息，这是预测模拟的一个重要方面。我们没有采用特定的ROM方法，因此特别吸引人的是，该方法适用于通过数据驱动方法获得的二次双线性模型，其中ROM稳定性（按定义）不能从完整的ROM中导出。订单模型。Burgers 方程的线性二次高斯 (LQG) 平衡 ROM、FitzHugh--Nagumo 的适当正交分解 ROM 和 Burgers 方程的非侵入式 ROM 的数值结果证明了所提出方法的可扩展性和定量优势。