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arxiv: 2504.14407 · v2 · submitted 2025-04-19 · 📡 eess.SY · cs.SY· math.OC

Soft and Hard Scaled Relative Graphs for Nonlinear Feedback Stability

Chao Chen , Sei Zhen Khong , Rodolphe Sepulchre This is my paper

Pith reviewed 2026-05-22 17:57 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords scaled relative graphsnonlinear feedback stabilityinput-output stabilityincremental passivitygraphical stability criteriaunbounded operators
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The pith

Separation of soft or hard scaled relative graphs on the complex plane certifies input-output stability for nonlinear feedback interconnections.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that if the soft scaled relative graphs or hard scaled relative graphs of two open-loop systems remain separated on the complex plane, their feedback interconnection is input-output stable. This graphical test works for systems that may be unbounded and removes the chordal assumption required in earlier soft SRG results. The approach treats soft SRGs as capturing incremental positivity and hard SRGs as capturing incremental passivity, reconciling the two from a single geometric viewpoint. A reader would care because the method supplies a direct way to check stability by inspecting relative positions rather than solving differential equations or verifying sector conditions.

Core claim

The separation of soft SRGs or hard SRGs of two open-loop systems on the complex plane guarantees closed-loop stability. The analysis applies to possibly unbounded open-loop systems based on their hard SRGs and does not require the chordal assumption used in prior soft SRG results.

What carries the argument

Soft and hard scaled relative graphs (SRGs), which are sets in the complex plane that encode the incremental input-output behavior of each open-loop system so that non-overlapping regions imply stability of the interconnection.

If this is right

  • Stability can be certified for feedback loops containing unbounded operators by using hard SRGs.
  • The chordal assumption is no longer needed when applying soft SRG separation.
  • Incremental positivity and incremental passivity receive a unified geometric treatment.
  • Graphical checks become available for a broader class of nonlinear systems than sector or small-gain methods alone permit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same separation idea might be applied to other interconnection structures such as cascades or parallel sums once the corresponding SRG combination rules are derived.
  • Numerical approximation of SRGs for concrete nonlinear models could turn the test into a practical computational check.
  • The distinction between soft and hard SRGs suggests that hybrid stability certificates mixing positivity and passivity properties are possible without additional assumptions.

Load-bearing premise

The open-loop systems must possess well-defined soft or hard SRGs whose separation on the complex plane is enough to conclude stability of the closed loop.

What would settle it

Two concrete systems whose soft or hard SRGs are visibly separated on the complex plane but whose feedback interconnection produces an unbounded trajectory or fails to satisfy an input-output stability definition.

Figures

Figures reproduced from arXiv: 2504.14407 by Chao Chen, Rodolphe Sepulchre, Sei Zhen Khong.

Figure 1
Figure 1. Figure 1: An upper bound (the gray sectored region D) of the soft SRG of a strictly incrementally positive system with indices δ, ǫ > 0. D. Connections Between Soft and Hard SRGs For a causal bounded system P , it is known from [4, p. 200] that the incremental positivity over L2 in (3) and incremental passivity over L2e in (4) are equivalent. We have also seen in Section II that kP kI can be equivalently defined on … view at source ↗
read the original abstract

This article 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 SRGs or 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 based on their hard SRGs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces soft and hard scaled relative graphs (SRGs) as a graphical tool for input-output stability analysis of nonlinear feedback systems. It claims that separation of the soft SRGs or 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 (previously limited to bounded systems under a chordal assumption) by removing that assumption and extending the analysis to possibly unbounded open-loop systems via their hard SRGs, while distinguishing incremental positivity from incremental passivity.

Significance. If the central claims hold, the work supplies a unified graphical criterion that handles both bounded and unbounded operators without the chordal restriction, potentially broadening the applicability of SRG methods to a wider class of nonlinear systems in control theory. The explicit reconciliation of positivity and passivity perspectives is a constructive contribution.

major comments (1)
  1. [Abstract and §4] Abstract and §4 (main stability theorem): the claim that hard-SRG separation on the complex plane guarantees input-output stability for unbounded open-loop systems is load-bearing, yet the manuscript must explicitly show that separation implies well-posedness of the interconnection (i.e., existence and uniqueness of the closed-loop map) rather than assuming the map exists a priori. For operators whose graphs are not closed or whose domains are not dense, this step is not automatic and requires a separate argument or regularity condition.
minor comments (2)
  1. [§2] §2 (definitions): the distinction between soft and hard SRGs is introduced clearly, but the precise construction of the hard SRG for unbounded operators should include an explicit statement of the domain and range assumptions used in the subsequent interconnection lemmas.
  2. [Figure 2] Figure 2 and surrounding text: the complex-plane plots of SRG separation would benefit from an added annotation indicating the precise separation margin that is proved sufficient for stability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback and positive assessment of the manuscript's contributions. We address the major comment below and will revise the manuscript to strengthen the presentation of the main result.

read point-by-point responses

  1. Referee: [Abstract and §4] Abstract and §4 (main stability theorem): the claim that hard-SRG separation on the complex plane guarantees input-output stability for unbounded open-loop systems is load-bearing, yet the manuscript must explicitly show that separation implies well-posedness of the interconnection (i.e., existence and uniqueness of the closed-loop map) rather than assuming the map exists a priori. For operators whose graphs are not closed or whose domains are not dense, this step is not automatic and requires a separate argument or regularity condition.

    Authors: We thank the referee for highlighting this important point. The current statement of the main stability theorem in Section 4 (and the abstract) assumes that the feedback interconnection is well-posed when SRG separation is used to conclude input-output stability. We agree that this assumption should be made explicit and justified, particularly for unbounded operators where well-posedness is not automatic. In the revised manuscript we will add a supporting lemma (or remark) that provides sufficient regularity conditions—such as closedness of the operator graphs or density of the domains—under which hard-SRG separation implies existence and uniqueness of the closed-loop map. This addition will be placed before the main theorem and will not alter the core claims or proofs. revision: yes

Circularity Check

0 steps flagged

No significant circularity in SRG separation generalization

full rationale

The paper generalizes a prior soft SRG separation theorem (for bounded systems under chordal assumption) to hard SRGs for possibly unbounded open-loop systems, claiming that graphical separation on the complex plane guarantees closed-loop input-output stability. This extension is presented via new theorems on interconnection properties of soft/hard SRGs, without reducing the central claim to a self-definition, fitted parameter renamed as prediction, or unverified self-citation chain. The derivation builds on cited prior results but introduces independent content for the unbounded case and removal of the chordal assumption; no equations or steps in the provided abstract and description exhibit the specific reductions required for higher circularity scores. The analysis remains self-contained against external graphical stability benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The abstract does not introduce explicit free parameters or new physical entities. The SRG notions themselves are the central modeling device. Standard mathematical assumptions on well-posed interconnections and causality are implicitly required but not detailed here.

axioms (1)
  • domain assumption Open-loop systems admit well-defined soft or hard scaled relative graphs whose separation implies closed-loop input-output stability.
    This premise is invoked when the abstract claims that SRG separation guarantees stability for both bounded and unbounded cases.
invented entities (1)
  • Hard scaled relative graph (hard SRG) no independent evidence
    purpose: Graphical representation that captures incremental passivity for stability analysis of possibly unbounded systems.
    Introduced to extend the prior soft SRG framework; no independent falsifiable prediction outside the stability theorem is given in the abstract.

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Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Scaled Graph Containment for Feedback Stability: Soft-Hard Equivalence and Conic Regions

    math.OC 2026-04 unverdicted novelty 7.0

    Soft-hard equivalence in circular scaled graph containment bypasses computational constraints for feedback stability, while hyperbolically convex conics yield tighter bounds for nonsymmetric cases.

  2. Scaled Relative Graphs in Normed Spaces

    math.OC 2026-04 unverdicted novelty 7.0

    Scaled relative graphs are extended to normed spaces via directional pairings from regular pairings, yielding geometric containment tests for contraction and monotonicity.

  3. Scaled Graph Bounding Techniques for Reset Systems

    math.OC 2026-05 unverdicted novelty 6.0

    Derives techniques to over-bound scaled graphs of reset systems using piecewise quadratic storage functions and identifies a fundamental limitation of quadratic dissipativity-based approximations via sampling.

  4. Symmetry Is Almost All You Need: Robust Stability with Uncertainty Induced by Symmetric SRG Regions

    eess.SY 2026-04 unverdicted novelty 6.0

    Mirror symmetry of SRG uncertainty regions about the theta-axis gives necessary and sufficient conditions for robust nonsingularity and stability of LTI systems via the Davis-Wielandt shell.

  5. Computable Characterisations of Scaled Relative Graphs of Closed Operators

    eess.SY 2025-11 unverdicted novelty 6.0

    Exact and computable constructions of Scaled Relative Graphs for closed linear operators are given via maximum and minimum gain computations, with a Bounded Real Lemma route for state-space models.

Reference graph

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