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Soft and Hard Scaled Relative Graphs for Nonlinear Feedback Stability

Chao Chen, Sei Zhen Khong, Rodolphe Sepulchre

TL;DR

The paper develops a graphical framework for nonlinear feedback stability based on scaled relative graphs (SRGs), distinguishing soft SRGs for incremental positivity and hard SRGs for incremental passivity. It proves that separation of the SRGs of two open-loop systems in the complex plane guarantees closed-loop stability, and it extends prior chordal SRG results by removing chordal assumptions and allowing unbounded open-loop systems. Key contributions include two main SRG-separation theorems—hard-SRG and soft-SRG—within an $\mathcal{L}_2$ and $\mathcal{L}_{2e}$ setting, respectively, with the hard SRG offering a direct, non-homotopic stability proof. The work connects SRG separation to classical incremental positivity/passivity results, providing a practical and visual criterion for nonlinear feedback stability and paving the way for robust and uncertain-system extensions.

Abstract

This paper presents input-output stability analysis of nonlinear feedback systems based on the notion of soft and hard scaled relative graphs (SRGs). The soft and hard SRGs acknowledge the distinction between incremental positivity and incremental passivity and reconcile them from a graphical perspective. The essence of our proposed analysis is that the separation of soft/hard SRGs of two open-loop systems on the complex plane guarantees closed-loop stability. The main results generalize an existing soft SRG separation theorem for bounded open-loop systems which was proved based on interconnection properties of soft SRGs under a chordal assumption. By comparison, our analysis does not require this chordal assumption and applies to possibly unbounded open-loop systems.

Soft and Hard Scaled Relative Graphs for Nonlinear Feedback Stability

TL;DR

The paper develops a graphical framework for nonlinear feedback stability based on scaled relative graphs (SRGs), distinguishing soft SRGs for incremental positivity and hard SRGs for incremental passivity. It proves that separation of the SRGs of two open-loop systems in the complex plane guarantees closed-loop stability, and it extends prior chordal SRG results by removing chordal assumptions and allowing unbounded open-loop systems. Key contributions include two main SRG-separation theorems—hard-SRG and soft-SRG—within an and setting, respectively, with the hard SRG offering a direct, non-homotopic stability proof. The work connects SRG separation to classical incremental positivity/passivity results, providing a practical and visual criterion for nonlinear feedback stability and paving the way for robust and uncertain-system extensions.

Abstract

This paper presents input-output stability analysis of nonlinear feedback systems based on the notion of soft and hard scaled relative graphs (SRGs). The soft and hard SRGs acknowledge the distinction between incremental positivity and incremental passivity and reconcile them from a graphical perspective. The essence of our proposed analysis is that the separation of soft/hard SRGs of two open-loop systems on the complex plane guarantees closed-loop stability. The main results generalize an existing soft SRG separation theorem for bounded open-loop systems which was proved based on interconnection properties of soft SRGs under a chordal assumption. By comparison, our analysis does not require this chordal assumption and applies to possibly unbounded open-loop systems.

Paper Structure

This paper contains 15 sections, 6 theorems, 67 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Basic Notation
  4. Signals and Systems
  5. Soft and Hard Scaled Relative Graphs
  6. Soft SRGs
  7. Hard SRGs
  8. Characterizations of Incremental Positivity and Incremental Passivity via Soft and Hard SRGs
  9. Connections Between Soft and Hard SRGs
  10. Main Results: Soft and Hard SRG Separation
  11. Hard SRG Separation
  12. Soft SRG Separation
  13. Graphical Characterizations of Incremental Positivity Theorems and Incremental Passivity Theorems
  14. Relations with Existing Soft SRG Separation
  15. Conclusion

Key Result

Proposition 1

For a strictly incrementally positive $\boldsymbol{P}$ defined by eq: l2_inc_spassive_def, it holds that $\mathrm{SRG}(\boldsymbol{P}) \subset \mathcal{D}$. Similarly, for a strictly incrementally passive $\boldsymbol{P}$ defined by eq: l2e_inc_spassive_def, we have $\mathrm{SRG}_e(\boldsymbol{P}) \

Figures (2)

  • Figure 1: An upper bound (the gray sectored region $\mathcal{D}$) of the soft SRG of a strictly incrementally positive system with indices $\delta, \epsilon>0$.
  • Figure 2: A feedback system $\hbox{${\boldsymbol{P}}\,\#\,\boldsymbol{C}$}$.

Theorems & Definitions (16)

  • Definition 1: Open-loop stability
  • Definition 2: Positivity and passivity
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Definition 3: Feedback well-posedness
  • Definition 4: Feedback stability
  • Theorem 1: Hard SRG Separation
  • proof
  • ...and 6 more