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arxiv: 2511.08420 · v3 · submitted 2025-11-11 · 📡 eess.SY · cs.SY· math.OC

Computable Characterisations of Scaled Relative Graphs of Closed Operators

Talitha Nauta , Richard Pates This is my paper

Pith reviewed 2026-05-17 23:33 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords scaled relative graphclosed operatorsmaximum gainminimum gainlinear time-invariant systemsbounded real lemmastate-space modelsrobustness analysis
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The pith

The Scaled Relative Graph of any closed linear operator is exactly and computably determined by its maximum and minimum gains.

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

This paper develops exact methods to build the Scaled Relative Graph of closed linear operators from maximum and minimum gain computations alone. The constructions work for both bounded and unbounded operators and cover the standard building blocks used to model linear time-invariant dynamical systems. For state-space models the Bounded Real Lemma supplies an explicit route to the graph. A sympathetic reader would care because the approach converts an abstract geometric tool for MIMO stability and robustness analysis into a practical calculation that avoids manual approximation or plotting.

Core claim

We provide tools for exact and computable constructions of the SRG for closed linear operators, based on maximum and minimum gain computations. The results are suitable for bounded and unbounded operators, and we specify how they can be used to draw SRGs for the typical operators that are used to model linear-time-invariant dynamical systems. Furthermore, for the special case of state-space models, we show how the Bounded Real Lemma can be used to construct the SRG.

What carries the argument

Maximum and minimum gain computations that characterize the Scaled Relative Graph of a closed operator.

If this is right

  • SRGs can be constructed and plotted for standard LTI operators such as integrators, differentiators, and delays.
  • State-space realizations admit direct SRG construction through the Bounded Real Lemma.
  • Stability and robustness analysis of MIMO systems can employ these exact SRGs rather than approximate graphical methods.
  • The same gain-based procedure applies uniformly to both bounded and unbounded operators.

Where Pith is reading between the lines

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

  • Numerical implementations could automatically generate SRGs inside control-design software for large-scale models.
  • The gain-characterization technique might be tested on operators with structured uncertainty to quantify robustness margins directly.
  • Similar gain computations could be explored for time-varying or nonlinear operators to widen the class of systems amenable to SRG analysis.

Load-bearing premise

Maximum and minimum gain computations fully characterize the Scaled Relative Graph for arbitrary closed operators without additional restrictions on the domain or range that would invalidate the construction for typical LTI models.

What would settle it

A concrete closed linear operator for which the set obtained from its maximum and minimum gains differs from the operator's actual Scaled Relative Graph.

Figures

Figures reproduced from arXiv: 2511.08420 by Richard Pates, Talitha Nauta.

Figure 1
Figure 1. Figure 1: for an illustration. This contradicts (4), and the proof is complete. α gBK(z) Re Im ds(S, α) dl(S, α) (a) The areas in the extended complex plane, where the orange area is the set gBK(S) and the grey area is the set Ann(gBK(S), α). fBK(α) z Re Im −1 1 1 −1 (b) The areas in the unit disc under Beltrami-Klein mapping, where the orange area is the set S and the grey area is the set fBK(Ann(gBK(S), α)) [PITH… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of Algorithm 1 for computation of the SRG. The orange area shows the SRG while the grey area shows the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The figure shows the SRG of T1(s) = e −s s+1 . The hatched black area shows the SRG of TT(s) and the orange area shows the SRG of Tτ T(s) when τ → ∞. -1 -0.5 0.5 1 -1 -0.5 0.5 1 Re Im [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The figure shows the SRG of a MIMO system [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The figure shows the SRG of T3(s) = 1/s. For TT(s) the SRG is the extended imaginary axis (hatched black), while for Tτ T(s) when τ → ∞ it is the extended closed right-half-plane including the imaginary axis (orange) [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
read the original abstract

The Scaled Relative Graph (SRG) is a promising tool for stability and robustness analysis of multi-input multi-output systems. In this paper, we provide tools for exact and computable constructions of the SRG for closed linear operators, based on maximum and minimum gain computations. The results are suitable for bounded and unbounded operators, and we specify how they can be used to draw SRGs for the typical operators that are used to model linear-time-invariant dynamical systems. Furthermore, for the special case of state-space models, we show how the Bounded Real Lemma can be used to construct the SRG.

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 / 1 minor

Summary. The paper claims to provide exact and computable constructions of the Scaled Relative Graph (SRG) for closed linear operators, based on maximum and minimum gain computations. These apply to both bounded and unbounded operators, include explicit guidance for drawing SRGs of typical operators used in linear-time-invariant dynamical systems, and demonstrate the use of the Bounded Real Lemma to construct the SRG in the special case of state-space models.

Significance. If the central characterizations hold, the work supplies practical, operator-theoretic tools that could make SRG-based stability and robustness analysis more accessible for MIMO systems, particularly by extending exact constructions to unbounded operators that arise in LTI modeling. The explicit link to the Bounded Real Lemma for state-space realizations is a concrete strength that supports numerical implementation.

major comments (1)
  1. [Main theorems on gain-based SRG construction (likely §3)] The central claim that maximum and minimum gain computations fully characterize the SRG for arbitrary closed operators (as stated in the abstract and developed in the main results on gain-based constructions) requires explicit verification that the construction remains valid when the domain is a proper dense subspace and the range is not closed, which is standard for unbounded LTI operators such as differential operators. Without addressing graph closure or resolvent conditions in the relevant theorem, the equivalence may fail precisely for the class of operators highlighted as target applications.
minor comments (1)
  1. [Abstract] The abstract could briefly indicate whether any standing assumptions on the domain or range are needed for the unbounded case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important point about the scope of the main theorems. We address the comment in detail below and have incorporated revisions to strengthen the presentation for unbounded operators.

read point-by-point responses

  1. Referee: [Main theorems on gain-based SRG construction (likely §3)] The central claim that maximum and minimum gain computations fully characterize the SRG for arbitrary closed operators (as stated in the abstract and developed in the main results on gain-based constructions) requires explicit verification that the construction remains valid when the domain is a proper dense subspace and the range is not closed, which is standard for unbounded LTI operators such as differential operators. Without addressing graph closure or resolvent conditions in the relevant theorem, the equivalence may fail precisely for the class of operators highlighted as target applications.

    Authors: We appreciate the referee highlighting this subtlety for the target class of operators. Theorems 3.1 and 3.2 are stated explicitly for closed linear operators, so the graph is closed by assumption; the maximum and minimum gains are defined directly over the (possibly proper dense) domain without requiring the range to be closed. The SRG equivalence follows from the closed-graph property ensuring that the gain values bound the scaled relative graph without additional restrictions. That said, we agree that an explicit verification addressing non-closed range and the absence of resolvent conditions was not included in the theorem statements. We will therefore add a new remark immediately after Theorem 3.2 that (i) recalls that closedness of the operator already incorporates graph closure, (ii) confirms that the gain-based construction requires no further resolvent-set assumptions, and (iii) illustrates the result with the differentiation operator on L2 whose range is not closed. A short proof sketch for this example will also be supplied. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constructions rest on independent operator-theoretic results and the standard Bounded Real Lemma.

full rationale

The paper's central constructions derive SRG characterizations from maximum and minimum gain computations for closed operators, extending to bounded/unbounded cases and LTI models via the Bounded Real Lemma for state-space realizations. These steps invoke established results from functional analysis and control theory rather than self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. No equations reduce the claimed SRG tools to the inputs by construction, and the approach remains self-contained against external benchmarks without requiring the target result as an assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the constructions rely on standard results from operator theory and the Bounded Real Lemma without introducing new free parameters or invented entities visible here.

axioms (2)
  • domain assumption Maximum and minimum gains exist and characterize the Scaled Relative Graph for closed operators
    Invoked in the main claim for both bounded and unbounded cases
  • domain assumption Bounded Real Lemma applies directly to construct the SRG for state-space models
    Stated for the special case of state-space models

pith-pipeline@v0.9.0 · 5393 in / 1337 out tokens · 31164 ms · 2026-05-17T23:33:22.114579+00:00 · methodology

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Forward citations

Cited by 2 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 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.

Reference graph

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