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

Xiaokan Yang, Ding Zhang, Wei Chen, Li Qiu

TL;DR

This work develops a θ-symmetric scaled relative graph (SRG) to fuse gain and refined phase information for graphical stability analysis. It defines a generalized angle between complex vectors and shows that θ-symmetric SRG reduces to the standard SRG when θ=0, while offering tighter phase capture and a MIMO Nyquist-like frequency plot. A crucial submultiplicative property enables a stability criterion for cyclic cascaded interconnections: if there exist real α_i(ω) with -1 not in the product of θ-symmetric SRGs of the blocks, the cyclic interconnection is stable, with a corollary recovering standard SRG results. Over-approximations via θ-arc hulls and associated matrix sets provide practical, tractable conditions, and two numerical examples illustrate both the gain- and phase-based advantages and the method’s applicability to MIMO cascades. Overall, the θ-symmetric SRG subsumes existing results, reduces conservatism, and supports extensions to broader networked systems.

Abstract

In this paper, we utilize a variant of the scaled relative graph (SRG), referred to as the $θ$-symmetric SRG, to develop a graphical stability criterion for the feedback interconnection of a cascade of systems. A crucial submultiplicative property of $θ$-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 $θ$-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 $θ$-symmetric SRG can be reduced exactly to the scalar itself, whereas the standard SRG appears to be a conjugate pair. Consequently, the frequency-wise $θ$-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 $θ$-symmetric SRG.

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

TL;DR

This work develops a θ-symmetric scaled relative graph (SRG) to fuse gain and refined phase information for graphical stability analysis. It defines a generalized angle between complex vectors and shows that θ-symmetric SRG reduces to the standard SRG when θ=0, while offering tighter phase capture and a MIMO Nyquist-like frequency plot. A crucial submultiplicative property enables a stability criterion for cyclic cascaded interconnections: if there exist real α_i(ω) with -1 not in the product of θ-symmetric SRGs of the blocks, the cyclic interconnection is stable, with a corollary recovering standard SRG results. Over-approximations via θ-arc hulls and associated matrix sets provide practical, tractable conditions, and two numerical examples illustrate both the gain- and phase-based advantages and the method’s applicability to MIMO cascades. Overall, the θ-symmetric SRG subsumes existing results, reduces conservatism, and supports extensions to broader networked systems.

Abstract

In this paper, we utilize a variant of the scaled relative graph (SRG), referred to as the -symmetric SRG, to develop a graphical stability criterion for the feedback interconnection of a cascade of systems. A crucial submultiplicative property of -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 -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 -symmetric SRG can be reduced exactly to the scalar itself, whereas the standard SRG appears to be a conjugate pair. Consequently, the frequency-wise -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 -symmetric SRG.

Paper Structure

This paper contains 11 sections, 16 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on SRG
  3. Definition of SRG
  4. Properties of SRG
  5. $\theta$-Symmetric SRG of Complex Matrices
  6. Definition and Properties
  7. $\theta$-Symmetric SRG of Cascaded Product versus Product of Corresponding $\theta$-Symmetric SRGs
  8. Over-approximations of Matrices via $\theta$-Symmetric SRG
  9. Stability Analysis of Cyclic Cascaded Systems via $\theta$-symmetric SRG
  10. Numerical Examples
  11. Conclusions

Key Result

Lemma 1

Let $x,y,z\in \mathbb{C}^{n}$ be nonzero vectors, then

Figures (4)

  • Figure 1: Feedback interconnection of a cascade of systems.
  • Figure 2: An illustration of $\mathrm{SRG}(C)$ (gray region) and $\mathrm{SRG}_\theta(C)$ (green region) with $\theta = 50^\circ$, where the red points are the eigenvalues of $C$.
  • Figure 3: An illustration of $\mathrm{SRG}_\theta(C)$ (cyan-green region) and its over-approximation $\mathscr{S}_\theta(C)$ (purple region) with $\theta = 67^\circ$, where the red points are the eigenvalues of $C$.
  • Figure 4: An illustration of the frequency-wise $\theta$-symmetric SRG of $G(j\omega)$ in \ref{['eq:system_example']}, where the red curves denote the eigenloci of $G(j\omega)$.

Theorems & Definitions (33)

  • Example 1
  • Lemma 2
  • Lemma 3: Ryu2022SRGChaffey2023Graphical
  • Example 2
  • Example 3
  • Remark 1
  • Lemma 4
  • proof
  • Definition 1: $\theta$-chord property
  • Proposition 1
  • ...and 23 more