The $θ$-Symmetric SRG with Applications to Stability of Cactus Dynamic Networks
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.