AI 中文总结
本文提出θ对称SRG，推导其代数性质并建立与范数最小化的联系，构建多环仙人掌网络统一稳定性框架，其保守性更低且几何解释更直观。
AI 中文摘要
本文系统研究了缩放相对图（SRG）的一种变体，称为θ对称SRG，并将其应用于仙人掌网络的稳定性分析。与先前的SRG定义相比，θ对称SRG能够刻画相位超前和滞后行为，是经典奈奎斯特图更自然的多变量扩展。我们首先分别分析θ对称SRG的增益和相位特性，建立θ分段相位与范数最小化问题的联系，该联系使得通过半定规划计算θ分段相位成为可能。我们进一步推导了θ对称SRG的次乘性和次加性性质，这些代数性质对于确定乘积型和和型回差矩阵的非奇异性至关重要，这两种矩阵在拓扑上对应于仙人掌网络的循环和并行反馈极端情况。以循环互联为基本出发点，我们建立了其鲁棒稳定性的充要条件，结合并行情况，合成了通用多环仙人掌网络的统一稳定性框架。与现有部分方法相比，θ对称SRG框架保守性更低，能为系统行为提供更直观的几何解释，文中包含多个示例以证明所提方法的有效性。
英文摘要
In this paper, we systematically study a variant of the scaled relative graph (SRG), referred to as the $θ$-symmetric SRG, and apply it to the stability analysis of cactus networks. Compared with the previous SRG definition, the $θ$-symmetric SRG enables the characterization of phase lead and lag behaviors, and serves as a more natural multivariable extension of the classical Nyquist plot. We first analyze the gain and phase aspects of $θ$-symmetric SRG separately and build a connection between $θ$-segmental phase and a norm minimization problem. This connection makes it possible to compute $θ$-segmental phase via semidefinite programming. We further derive the submultiplicative and subadditive properties of $θ$-symmetric SRG. These algebraic properties are crucial to determine the nonsingularity of product-type and sum-type return difference matrices, which topologically correspond to the cyclic and parallel-feedback extreme cases of cactus networks. By taking the cyclic interconnection as the fundamental starting point, we establish necessary and sufficient conditions for its robust stability. Integrating this with the parallel case, we synthesize a unified stability framework for general multi-loop cactus networks. The $θ$-symmetric SRG framework is less conservative and provides a more intuitive geometric interpretation of system behaviors compared with some existing approaches. Several examples are included to demonstrate the effectiveness of the proposed methods.
Comments16 pages, 10 figures
