New θ-Symmetric SRG framework simplifies stability checks for cactus networks

In a new arXiv paper (2608.12591), Xiaokan Yang, Wei Chen, and Li Qiu from the systems and control community systematically develop the θ-symmetric scaled relative graph (SRG), a variant that overcomes key limitations of previous SRG definitions. Standard SRGs are powerful for analyzing feedback loops but struggle to represent phase lead and lag behaviors. The θ-symmetric SRG solves this by separating gain and phase analysis into a unified geometric object, effectively serving as a more natural multivariable extension of the classical Nyquist plot. The authors demonstrate that the θ-segmental phase can be computed via semidefinite programming, which opens the door to efficient numerical algorithms for robustness verification.

A major contribution is the derivation of submultiplicative and subadditive properties for θ-symmetric SRGs. These algebraic properties are essential for determining 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. Taking the cyclic interconnection as the fundamental starting point, the team establishes necessary and sufficient conditions for robust stability. Integrating these results with the parallel case yields a unified stability framework for general multi-loop cactus networks—topologies common in power grids, communication systems, and distributed control. The proposed framework is less conservative than existing approaches and provides a more intuitive geometric interpretation of system behavior. With 16 pages and 10 figures, the paper includes several examples confirming the effectiveness of the method.