In a recent work [Manucci, Unger, ArXiv e-print 2404.10511, 2024], the authors propose using two generalized Lyapunov equations (GLEs) to derive a balancing-based model order reduction~(MOR) method for a general class of switched differential-algebraic equations (DAEs). This work explains why these GLEs provide solutions suitable for MOR by showing that the image set of the solutions of the two GLEs always encloses the reachable and observable set of a suitably defined switched system with the same input to output map of the switched DAE system.

翻译：在近期工作[Manucci, Unger, ArXiv e-print 2404.10511, 2024]中，作者提出利用两个广义Lyapunov方程，为一类通用的切换微分代数方程系统推导出基于平衡截断的模型降阶方法。本文通过证明这两个广义Ly雅普诺夫方程解的像集始终包含一个具有相同输入输出映射的切换系统的可达集与可观集，阐明了为何这些广义Lyapunov方程能够为模型降阶提供合适的解。