A Cascade of Systems and the Product of Their $θ$-Symmetric Scaled Relative Graphs

In this paper, we utilize a variant of the scaled relative graph (SRG), referred to as the $\theta$-symmetric SRG, to develop a graphical stability criterion for the feedback interconnection of a cascade of systems. A crucial submultiplicative property of $\theta$-symmetric SRG is established, enabling it to handle cyclic interconnections for which conventional graph separation methods are not applicable. By integrating both gain and refined phase information, the $\theta$-symmetric SRG provides a unified graphical characterization of the system, which better captures system properties and yields less conservative results. In the scalar case, the $\theta$-symmetric SRG can be reduced exactly to the scalar itself, whereas the standard SRG appears to be a conjugate pair. Consequently, the frequency-wise $\theta$-symmetric SRG is more suitable than the standard SRG as a multi-input multi-output extension of the classical Nyquist plot. Illustrative examples are included to demonstrate the effectiveness of the $\theta$-symmetric SRG.