对角线不等式解的块对角矩阵及对角主导性的推广
Block-Diagonal Solutions to Lyapunov Inequalities and Generalisations of Diagonal Dominance
AI总结：
本文研究了块对角矩阵对角主导性的扩展，展示了如何通过线性代数工具求解Lyapunov不等式，并通过数值示例说明了方法的优势与局限。
摘要：
对角主导矩阵在系统与控制理论中有广泛应用。具有标定对角主导漂移矩阵的线性动态系统，包括稳定的正系统，允许可扩展的稳定性分析。例如，已知此类系统的Lyapunov不等式允许对角解。本文提出将标定对角主导性扩展到分块分块矩阵。我们证明我们的定义描述了允许Lyapunov不等式块对角解的矩阵，并且这些解可通过线性代数工具计算。我们还展示了在某些情况下，Lyapunov不等式可以解耦为一组低维线性矩阵不等式，从而导致改进的可扩展性。最后，通过数值示例说明了我们结果的优势和局限。
英文摘要：
Diagonally dominant matrices have many applications in systems and control theory. Linear dynamical systems with scaled diagonally dominant drift matrices, which include stable positive systems, allow for scalable stability analysis. For example, it is known that Lyapunov inequalities for this class of systems admit diagonal solutions. In this paper, we present an extension of scaled diagonally dominance to block partitioned matrices. We show that our definition describes matrices admitting block-diagonal solutions to Lyapunov inequalities and that these solutions can be computed using linear algebraic tools. We also show how in some cases the Lyapunov inequalities can be decoupled into a set of lower dimensional linear matrix inequalities, thus leading to improved scalability. We conclude by illustrating some advantages and limitations of our results with numerical examples.
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