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Mixed Small Gain and Phase Theorem: A new view using Scale Relative Graphs

Eder Baron-Prada, Adolfo Anta, Alberto Padoan, Florian Dörfler

TL;DR

This paper removes the sectoriality limitation of the small phase theorem by introducing Scaled Relative Graphs (SRGs) as a versatile, set-based tool for LTI stability analysis. It develops a graphical SRG-based stability condition and a sectoriality-free small-phase theorem, enabling robust phase analysis for MIMO systems. The authors define maximum gain and maximum phase from SRGs and show how a mixed gain/phase criterion can certify stability in feedback, with frequency-by-frequency SRG considerations. Through numerical examples, the work demonstrates that SRG-based methods can certify stability in cases where traditional small-phase or small-gain tests fail, and it clarifies the trade-offs between exact SRG-based analysis and practical, conservative approximations. The findings broaden the practical toolkit for feedback stability in complex systems and point to future directions in decentralizing SRG-based criteria.

Abstract

We introduce a novel approach to feedback stability analysis for linear time-invariant (LTI) systems, overcoming the limitations of the sectoriality assumption in the small phase theorem. While phase analysis for single-input single-output (SISO) systems is well-established, multi-input multi-output (MIMO) systems lack a comprehensive phase analysis until recent advances introduced with the small-phase theorem. A limitation of the small-phase theorem is the sectorial condition, which states that an operator's eigenvalues must lie within a specified angle sector of the complex plane. We propose a framework based on Scaled Relative Graphs (SRGs) to remove this assumption. We derive two main results: a graphical set-based stability condition using SRGs and a small-phase theorem with no sectorial assumption. These results broaden the scope of phase analysis and feedback stability for MIMO systems.

Mixed Small Gain and Phase Theorem: A new view using Scale Relative Graphs

TL;DR

This paper removes the sectoriality limitation of the small phase theorem by introducing Scaled Relative Graphs (SRGs) as a versatile, set-based tool for LTI stability analysis. It develops a graphical SRG-based stability condition and a sectoriality-free small-phase theorem, enabling robust phase analysis for MIMO systems. The authors define maximum gain and maximum phase from SRGs and show how a mixed gain/phase criterion can certify stability in feedback, with frequency-by-frequency SRG considerations. Through numerical examples, the work demonstrates that SRG-based methods can certify stability in cases where traditional small-phase or small-gain tests fail, and it clarifies the trade-offs between exact SRG-based analysis and practical, conservative approximations. The findings broaden the practical toolkit for feedback stability in complex systems and point to future directions in decentralizing SRG-based criteria.

Abstract

We introduce a novel approach to feedback stability analysis for linear time-invariant (LTI) systems, overcoming the limitations of the sectoriality assumption in the small phase theorem. While phase analysis for single-input single-output (SISO) systems is well-established, multi-input multi-output (MIMO) systems lack a comprehensive phase analysis until recent advances introduced with the small-phase theorem. A limitation of the small-phase theorem is the sectorial condition, which states that an operator's eigenvalues must lie within a specified angle sector of the complex plane. We propose a framework based on Scaled Relative Graphs (SRGs) to remove this assumption. We derive two main results: a graphical set-based stability condition using SRGs and a small-phase theorem with no sectorial assumption. These results broaden the scope of phase analysis and feedback stability for MIMO systems.

Paper Structure

This paper contains 22 sections, 6 theorems, 23 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Signal Spaces
  4. Transfer Functions and Stability of LTI Systems
  5. Review: Small phase theorem
  6. Classification of Sectorial Operators
  7. The Small Phase Theorem
  8. Scaled Relative Graphs (SRGs)
  9. Scaled Relative Graphs of Square Matrices
  10. Stability Conditions based on SRGs
  11. Mixed Gain and Phase Theorem
  12. Maximum gain and phase
  13. Maximum Gain
  14. Maximum phase
  15. Small phase theorem using SRGs
  16. ...and 7 more sections

Key Result

Theorem 1

(Small phase theoremChen2019) Assume $H_1(s) \in \mathcal{RH}_\infty$ and $H_{2}(s) \in \mathcal{RH}_\infty$ are connected in a feedback loop as shown in Fig. fig:fb. If for each $s=j\omega$, with $\omega \in [0, \infty)$, the following holds: then, the closed-loop system is stable.

Figures (7)

  • Figure 1: (a) $W(A)$ of a sectorial operator $A$ with supporting angles ${\alpha}_{\max}(A)$ and ${\alpha}_{\min}(A)$. (b) $W(B)$ of a quasi-sectorial operator $B$ and $W(C)$ of a semi-sectorial operator $C$, respectively. (c) $W(D)$ of a non-sectorial operator $D$.
  • Figure 2: Feedback interconnection between $H_1(s)$ and $H_2(s)$.
  • Figure 3: Unitary feedback connection of $H(s)$.
  • Figure 4: Comparison between SRG-based phase calculation and phase calculation using sectorial properties.
  • Figure 5: (a) Numerical range of $H_1(j\omega)$ in gray for $\omega=0.1$ rad/s (b) $\operatorname{SRG}(H_1(j\omega))$, with $\hat{\alpha}_{\max} (H_1(j\omega))$ in blue and $\hat{\alpha}_{\min}(H_{1}(j\omega))$ in red with $\omega=0.1$ rad/s.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Remark 1: Over-approximation via right-arc property
  • Theorem 4
  • Theorem 5