On phase in scaled graphs

Alexander Lanzon; Chao Chen; Koen Scheres; Sebastiaan van den Eijnden; Thomas Chaffey

REVIEW 3 major objections 3 minor 26 references

On phase in scaled graphs

T0 review · 3 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The signed scaled graph, using Hilbert-transform phase, separates phase lead from phase lag in nonlinear feedback systems and yields a stability test that is never more conservative than the original scaled-graph test.

desk verdict The signed scaled graph is a genuinely useful new object, but the main stability theorem is not proven as stated because the proof analyzes a different separation condition than the one in the theorem. read the letter →

arxiv 2504.21448 v2 pith:PB75LLLE submitted 2025-04-30 eess.SY cs.SY

Sebastiaan van den Eijnden , Chao Chen , Koen Scheres , Thomas Chaffey , Alexander Lanzon This is my paper · ORCID

classification eess.SYcs.SY MSC 93C1093D2593C80

keywords signedscaledgraphHilberttransformphaseleadandlagnonlinearfeedbackstabilitypassivitynegativeimaginarysystemsgraphicaltest

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes changing the phase notion inside the scaled graph (SG), a graphical tool that plots gain and phase of nonlinear input-output trajectories like a Nyquist plot. The SG's phase is an unsigned singular angle, so a lead filter and a lag filter with the same gain profile look identical; the paper's signed scaled graph (SSG) replaces that angle with a signed phase computed from the Hilbert transform of the input, which carries the sign of the imaginary part of the frequency response in the linear case. The main result (Theorem 2) says negative feedback is finite-gain stable whenever the SSG of the forward system and the inverse SSG of the scaled negative feedback system are separated by a positive distance, for all gain scalings, except for a technical set of inputs. This recovers passivity and negative-imaginary interconnection results as special cases and, when applied to a lead-versus-lag example, reproduces the Nyquist stability boundary without conservatism.

What carries the argument

The signed scaled graph (SSG): for $u,y$, gain $\rho = \|y\|/\|u\|$, unsigned singular angle $\theta = \arccos(\langle u,y\rangle/(\|u\|\|y\|))$, and signed phase $\phi = \operatorname{sgn}(\langle\!\langle\!\langle \hat{u}, E(y)\rangle\!\rangle\!\rangle)\,\theta$, with a two-valued convention when the Hilbert inner product vanishes. The Hilbert-inner-product sign replaces the unsigned singular angle, so $\operatorname{SSG}(H) \subset \operatorname{SG}(H)$ and $\operatorname{SSG}(H) \cup \operatorname{SSG}^*(H) = \operatorname{SG}(H)$. The proof of Theorem 2 partitions trajectory pairs into small-gain, large-gain, and small-phase cases, using the graph-separation distance to obtain uniform $\epsilon$ or $\delta$ constants in each case, then invokes a homotopy and well-posedness argument to convert the uniform bounds into a finite-gain estimate.

What would settle it

Construct two finite-gain stable systems whose signed graphs are always separated by, say, distance 1 for every $\tau \in (0,1]$, but along a sequence of inputs the equal-magnitude points have phase differences that shrink to zero while norms grow, so no uniform $\delta$ exists for the small-phase case. If such a pair were also a counterexample to finite-gain stability, Theorem 2's conclusion would fail; if stability still held, the proof's uniformity premise would still be refuted.

Watch

Extended reading notes

Core claim

The paper's central claim is that phase information in a nonlinear operator can be made signed by defining phase from the Hilbert transform: for input $u$ and output $y$, the signed phase is the unsigned singular angle multiplied by the sign of the cross-correlation between the Hilbert transform of $u$ and the output. The resulting SSG is no longer symmetric about the real axis, and a positive-distance separation condition between $\operatorname{SSG}(H_1)$ and $\operatorname{SSG}^\dagger(-\tau H_2)$ for all $\tau \in (0,1]$ implies finite-gain stability of the negative feedback loop for all $L_2$ inputs outside the exceptional set $W$. Because the signed graph distinguishes phase lead from phase lag, the test is never more conservative than Theorem 1's unsigned separation test, and it recovers the classical passivity theorem and a negative-imaginary interconnection result as special cases. The authors define SSG-negative imaginary systems through a Hilbert-transform inequality and state a corresponding positive-feedback stability theorem.

Load-bearing premise

The proof's case distinction relies on the assumption that a fixed positive distance between the two signed graph sets forces a uniform gap in magnitude for equal-angle pairs or a uniform gap in angle for equal-magnitude pairs, with the same gap size for every trajectory magnitude; the paper does not show the graph sets are compact or bounded away from zero and infinity, so this uniformity is not automatic.

Editorial extensions

If this is right

For LTI systems, the SSG separates the positive-frequency and negative-frequency halves of the Nyquist diagram, so a lead filter and a lag filter with the same gain curve no longer produce the same graphical object.
Any feedback pair that passes the original scaled-graph separation test also passes the SSG test, and some pairs that fail the original test pass the SSG test; stability certificates are therefore at least as sharp.
The strict-passivity-plus-passivity theorem follows from the SSG separation condition with $W$ empty, giving finite-gain stability for all square-integrable inputs.
SSG-negative imaginary systems, defined via a Hilbert-transform inequality rather than time derivatives, admit a positive-feedback stability theorem under a DC-gain product condition.
The graphical separation condition can be checked by plane geometry, so the theorem functions as a nonlinear analogue of the Nyquist test.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the uniform $\epsilon/\delta$ gap can be shown to follow from a fixed set distance under mild compactness assumptions, Theorem 2 would hold for all $L_2$ inputs, removing the exceptional set $W$; the paper itself suggests $W$ is technical.
Because the Hilbert transform is an $L_2$ isometry up to sign, the SSG inequality defining SSG-negative imaginary systems may connect to integral quadratic constraint analyses for nonlinear systems, giving a frequency-domain interpretation that derivative-based negative-imaginary definitions lack.
For multi-input/multi-output operators the Hilbert inner product is still well-defined componentwise, so the construction appears to extend beyond scalar or single-channel settings, though only scalar signals are treated here.
The signed-phase construction could be used for synthesis, e.g., shaping a nonlinear controller's SSG to lie in the right half plane, which the introduction flags as a motivation for distinguishing lead from lag.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

You should know about this paper: it defines the signed scaled graph (SSG), a variant of the scaled graph that uses Hilbert-transform-based phase to distinguish lead from lag. That is a real idea, and the paper does several good things with it. It shows SSG separates lead and lag filters that are identical under the ordinary scaled graph, recovers passivity results, and gives a scaled-graph route to negative imaginary interconnections. The motivating examples are clear and honestly worked.

The soft spot is Theorem 2, the main stability result. The stated hypothesis (16) is dist(SSG†(H1), SSG(-τH2)) ≥ r, with the dagger on H1. But Appendix A defines z1 in SSG(H1) and z2 in SSG†(-τH2) — that is, it proves separation for dist(SSG(H1), SSG†(-τH2)), the condition the paper notes only in the sentence "Theorem 2 still holds if condition (16) is replaced by..." That sentence is doing real work: the proof supports only the variant, not the advertised theorem. The two conditions are not equivalent; a simple scalar example shows the distance can be positive one way and zero the other. So Theorem 2 as stated lacks support.

There is also a secondary gap. Even for the corrected variant, the proof jumps from a fixed distance lower bound r to a uniform epsilon (and later delta) controlling magnitude or angle separation. That inference requires compactness or explicit magnitude bounds, and none are supplied. The SSG sets can be unbounded or have accumulation points at zero, so the uniform constant may not exist.

The definition of the exceptional set W is honest — the paper says it is a technical restriction. But W is effectively defined as the set where the proof's hardest case appears, which is a mild bootstrap. Not fatal by itself, but it compounds the main issue.

Overall: the conceptual contribution is solid and worth engaging with. The proof needs a major repair. I would send it to peer review because the idea is novel and the examples are valuable, but I would expect the referee to demand a corrected theorem statement and a complete proof. If I were citing it, I would cite the SSG definition and the lead/lag separation example, not Theorem 2 as it stands.

Referee Report

3 major / 3 minor

Summary. The paper introduces a 'signed scaled graph' (SSG) for nonlinear systems, replacing the unsigned singular angle of the scaled graph with a phase defined through the Hilbert transform of the input. The main stability claim, Theorem 2, states that if the SSG of the inverse of H1 and the SSG of -tau H2 are separated by a positive distance for all tau in (0,1], then the negative feedback interconnection is finite-gain stable for all inputs outside a set W. The paper also proposes SSG-based definitions of passive and SSG-negative-imaginary systems, and argues that the signed phase distinguishes phase-lead from phase-lag, recovering passivity and negative-imaginary interconnection results and reducing conservatism relative to the original scaled graph. The lead/lag example and Theorem 3 illustrate the intended advantage of the new construction.

Significance. If Theorem 2 were established, the SSG would be a valuable extension of scaled-graph theory: it would provide a graphical stability test that separates phase lead from phase lag, and would unify passivity and negative-imaginary results in a single framework. The paper is clearly written, the construction is self-contained, and the motivating examples are compelling. Lemma 1 and Theorem 3 are elementary and correct. However, the main stability theorem is not soundly proven as stated, and the proof contains a load-bearing mismatch between the hypothesis and the argument. The significance of the paper is therefore prospective rather than established in its current form.

major comments (3)

[Theorem 2 and Appendix A] The proof of Theorem 2 does not address the hypothesis (16). Condition (16) is dist(SSG†(H1), SSG(−τH2)) ≥ r, but the proof's opening paragraph takes z1 ∈ SSG(H1) and z2 ∈ SSG†(−τH2), and the inequalities (21) and (23) are built from the forward gain of H1 paired with the inverse gain of H2. This is the reciprocal separation condition dist(SSG(H1), SSG†(−τH2)) ≥ r, which the paper only notes as an alternative under which the theorem 'still holds.' No argument shows that (16) implies this reciprocal condition, and it is not equivalent for arbitrary sets: for a=0.05 and b=10, dist({1/a}, {τb : τ∈(0,1]}) ≥ 10 while dist({a}, {1/(τb) : τ∈(0,1]}) = 0.05. Thus Theorem 2 as stated lacks proof support.
[Appendix A, uniform-epsilon step] The claim that |z1−z2| ≥ r is 'equivalent' to a uniform dichotomy—equal-angle points separated in magnitude by at least ε, or equal-magnitude points separated in angle by at least ε, for a single ε > 0—is not justified. The sets SSG(H1) and SSG(−τH2) are not shown to be compact, nor to be bounded away from zero and infinity, and trajectories may have arbitrarily large or small gains. A positive distance between two noncompact sets does not imply a uniform threshold ε (or later δ) independent of the trajectory magnitudes. Since the constants c1,...,c4 in (22), (24), (29), and (32) depend on this ε or δ, the homotopy argument requires an additional hypothesis (for example, boundedness of the SSG sets away from 0 and ∞) or a different proof technique.
[Equation (15) and Theorem 2] The excluded set W is defined in (15) through conditions on the internal signals u1, u2, and y2, so the conclusion 'finite-gain stable for all inputs w ∈ L2 \ W' is not an easily checkable input-space statement: membership of w in W depends on the solution of the feedback equations and on the resulting trajectories, not on w alone. The paper notes that W is empty in some classical cases and that W ≠ L2 in Example 2, but it does not quantify how large W can be. This weakens the practical force of the theorem; if W is intended as a technical exception, it should be characterized more directly or the theorem should be reformulated with an explicit condition on the data of the problem.

minor comments (3)

[Appendix A, notation] In the proof of Theorem 2, the displayed expression for z2 ∈ SSG†(−τH2) uses the forward gain ||−τy2||/||u2||, which is the magnitude of a point in SSG(−τH2), not in its inverse. The notation should be aligned with the hypothesis of the theorem or explicitly corrected.
[Example 2] The statement that SSG(H1) and −SSG†(τHlead) do not intersect is argued from the Nyquist diagram bounding SG(H1). Since the SSG is defined for all L2 inputs, the example should clarify why broadband nonlinear trajectories cannot leave the claimed half-plane; otherwise the graphical conclusion is only heuristic.
[Theorem 4] In the proof of Theorem 4, the sentence asserting that the separation of SSG(H1) \ X and SSG†(H2) \ X 'also holds for all scalings τ∈(0,1]' is stated without argument. Because scaling by τ changes magnitudes, a short justification would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the graph-separation derivation is self-contained; the Appendix A hypothesis mismatch is a correctness gap, not a circular step.

full rationale

The central claim (Theorem 2) is derived from the explicit graph-separation condition (16) through a homotopy/small-gain/large-gain/angle-separation argument in Appendix A; it does not fit any parameter to data, rename an empirical observation, or invoke a uniqueness theorem from the authors' prior work. The Hilbert-transform phase construction cites [17] for the identity ⟨⟨⟨ŷ,E(u)⟩⟩⟩ = −⟨⟨⟨û,E(y)⟩⟩⟩ and for the idea of nonlinear phase, but this is not load-bearing circularity: equations (11) and (12) provide independent LTI motivation, and the identity is an elementary Hilbert-transform property rather than an unverified ansatz imported solely by self-citation. The set W in (15) is an explicit technical exclusion set defined by signal inner products, norms, and a sign condition; it is not a disguised definition of "inputs for which stability fails," and the proof establishes finite-gain bounds on its complement. Per the reviewing rule, I flag two correctness concerns that do not constitute circularity: (i) Appendix A's proof analyzes z1 ∈ SSG(H1) and z2 ∈ SSG†(−τH2), whereas the stated hypothesis (16) involves SSG†(H1) and SSG(−τH2), so the proof appears to support the "still holds if" variant rather than the theorem exactly as stated; and (ii) the passage from dist(·,·) ≥ r to a uniform ǫ in the two separation cases is not justified by compactness or explicit magnitude bounds. These are validity gaps, not equivalences between inputs and outputs, so they do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The central claim rests on standard L2/Hilbert transform machinery, standing assumptions on system stability and well-posedness, and two unproven proof ingredients: a uniform separation constant and the excludability of W. No numerical parameters are fitted to data.

assumptions (4)

domain assumption Systems are stable causal maps H: L2 -> L2 with H(0) = 0, and feedback interconnections are well-posed for all tau in (0,1].
Used throughout; these are the standing assumptions of Sections II-IV and Theorem 2.
standard math Hilbert transform and Plancherel identities (11)-(12) compute real and imaginary parts of LTI frequency response from L2 inner products.
Used to justify the signed phase definition and the LTI consistency of the SSG-negative imaginary characterization.
ad hoc to paper Graph separation dist >= r implies a uniform dichotomous separation of magnitudes or arguments with thresholds independent of trajectory magnitudes.
Assumed at the start of the Appendix A proof of Theorem 2; not proven and not stated in the theorem.
ad hoc to paper The excluded input set W, defined by internal signal conditions in (15), can be ignored without destroying the usefulness of the stability claim.
Theorem 2 is proved only for w in L2 \ W; W is not shown to be empty or small, and the paper states that removing the restriction is future work.

invented entities (2)

Signed scaled graph (SSG)
purpose: Assigns signed phase to input-output pairs of nonlinear systems so phase lead and lag can be distinguished.
It is a new mathematical object; no external falsifiable prediction is made beyond internal consistency with passivity and LTI Nyquist facts.
SSG-negative imaginary system
purpose: A new nonlinear negative-imaginary class defined through Hilbert-transformed signals rather than derivatives.
The definition is introduced to fit the SSG framework; LTI consistency is demonstrated, but no independent experiment or benchmark validates the class.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On phase in scaled graphs." pith.science (2026). https://pith.science/paper/PB75LLLE

@misc{pith2026250421448,
  author       = {Pith},
  title        = {Pith review of: On phase in scaled graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PB75LLLE}},
  note         = {Machine review of arXiv:2504.21448}
}

read the original abstract

The scaled graph has been introduced recently as a nonlinear extension of the classical Nyquist plot for linear time-invariant systems. In this paper, we introduce a modified definition for the scaled graph, termed the signed scaled graph (SSG), in which the phase component is characterized by making use of the Hilbert transform. Whereas the original definition of the scaled graph uses unsigned phase angles, the new definition has signed phase angles which ensures the possibility to differentiate between phase-lead and phase-lag properties in a system. Making such distinction is important from both an analysis and a synthesis perspective, and helps in providing tighter stability estimates of feedback interconnections. We show how the proposed SSG leads to intuitive characterizations of positive real and negative imaginary nonlinear systems, and present various interconnection results. We showcase the effectiveness of our results through several motivating examples.

Figures

Figures reproduced from arXiv: 2504.21448 by the authors.

**Figure 1.** Negative feedback interconnection. Theorem 1. Consider a pair of stable systems H1 : L2 → L2 and H2 : L2 → L2, and suppose that the feedback interconnection in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references · 24 canonical work pages

[1]

Skogestad and I

S. Skogestad and I. Postletwaite, Multivariable Feedback Control: Analysis and Design . John Wiley & Sons, 2005

work page 2005
[2]

K. J. Astrom and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers . Princeton, NJ: Princeton University Press, 2010

work page 2010
[3]

Scaled relative graphs: Non- expansive operators via 2D Euclidean geometry,

E. K. Ryu, R. Hannah, and W. Yin, “Scaled relative graphs: Non- expansive operators via 2D Euclidean geometry,” Mathematical Pro- gramming, vol. 194, pp. 569–619, 2022

work page 2022
[4]

Scaled relative g raphs for system analysis,

T. Chaffey, F. Forni and R. Sepulchre, “Scaled relative g raphs for system analysis,” in Proc. 60th IEEE Conference on Decision and Control, Austin, TX, USA, pp. 3166-3172, 2021

work page 2021
[5]

A rolled-off passivity theorem,

T. Chaffey, “A rolled-off passivity theorem,” Systems & Control Letters, vol. 162, 2022

work page 2022
[6]

Graphical Nonli near System Analysis,

T. Chaffey, F. Forni, and R. Sepulchre, “Graphical Nonli near System Analysis,” IEEE Transactions on Automatic Control , vol. 68, no. 10, pp. 6067-6081, 2023

work page 2023
[7]

Scaled graphs for reset control system analysis,

S.J.A.M. van den Eijnden, T. Chaffey, T. Oomen, and W.P .M .H. Heemels, “Scaled graphs for reset control system analysis, ” European Journal of Control , vol. 80, art. 101050, 2024

work page 2024
[8]

The scaled relative graph of a linear operator

R. Pates, “The scaled relative graph of a linear operator ”, 2021, arXiv:2106.05650

arXiv 2021

Show all 26 references

[9]

van der Schaft, L2-Gain and Passivity Techniques in Non-linear Control, Springer, 2017

A. van der Schaft, L2-Gain and Passivity Techniques in Non-linear Control, Springer, 2017

2017
[10]

Wielandt, Topics in the Analytic Theory of Matrices

H. Wielandt, Topics in the Analytic Theory of Matrices. Madison, WI: University of Wisconsin Lecture Notes, 1967

1967
[12]

Soft and hard scal ed relative graphs for nonlinear feedback stability

C. Chen, S. Z. Khong and R. Sepulchre, “Soft and hard scal ed relative graphs for nonlinear feedback stability”, Under review, 20 24
[13]

On the input-output stability of time-varyi ng non-linear feedback systems Part one: Conditions derived using concep ts of loop gain, conicity, and positivity,

G. Zames, “On the input-output stability of time-varyi ng non-linear feedback systems Part one: Conditions derived using concep ts of loop gain, conicity, and positivity,” IEEE Transactions on Automatic Control, vol. 11, no. 2, pp. 228-238, 1966

1966
[14]

C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties. Society for Industrial and Applied Mathematics, 2009

2009
[15]

Astrom and T

K. Astrom and T. Hagglund, PID Controllers: Theory, Design, and Tuning. ISA - The Instrumentation, Systems and Automation Society , 1995

1995
[16]

F. W. King, Hilbert Transforms. New Y ork, NY: Cambridge University Press, 2009

2009
[17]

Phase of non- linear Systems,

C. Chen, D Zhao, W. Chen, S. Z. Khong, and L. Qiu, “Phase of non- linear Systems,” 2021, arXiv:2012.00692

2021 arXiv
[18]

Khalil, Non-linear Systems (3rd ed.)

H.K. Khalil, Non-linear Systems (3rd ed.). Prentice Hall, Upper Saddle River, New Jersey, 2002

2002
[19]

Extendi ng negative imaginary systems theory to nonlinear systems,

A. G. Ghallab, M. A. Mabrok, and I. R. Petersen, “Extendi ng negative imaginary systems theory to nonlinear systems,” i n Proc. IEEE Conference on Decision and Control (CDC) , Miami, FL, pp. 2348-2353, 2018

2018
[20]

Stability analysis of intercon nected systems with “mixed

S. Patra and A. Lanzon, “Stability analysis of intercon nected systems with “mixed” negative-imaginary and small-gain propertie s,” IEEE Transactions on Automatic Control , vol. 56, no. 6, pp. 1395-1400, 2011

2011
[21]

Characterization of input- output neg- ative imaginary systems in a dissipative framework,

A. Lanzon and P . Bhowmick, “Characterization of input- output neg- ative imaginary systems in a dissipative framework,” in IEEE Trans- actions on Automatic Control , vol. 68, no. 2, pp. 959-974, 2023

2023
[22]

A frequency-domain ap proach to nonlinear negative imaginary systems analysis,

D. Zhao, C. Chen, and S. Z. Khong, “A frequency-domain ap proach to nonlinear negative imaginary systems analysis,” Automatica, vol. 146, p. 110604, 2022

2022
[23]

Stability robustness of a feedback inter- connection of systems with negative imaginary frequency re sponse,

A. Lanzon and I. R. Petersen, “Stability robustness of a feedback inter- connection of systems with negative imaginary frequency re sponse,” IEEE Transactions on Automatic Control , vol. 53, no. 4, pp. 1042- 1046, 2008

2008
[24]

System analysis via integ ral quadratic constraints,

A. Megretski and A. Rantzer, “System analysis via integ ral quadratic constraints,” IEEE Transactions on Automatic Control , vol. 42, no. 6, pp. 819-830, 1997

1997
[25]

On a larg e-gain theorem,

T. T. Georgiou, M. Khammash, and A. Megretski, “On a larg e-gain theorem,” Systems & Control Letters , vol. 32, no. 4, pp. 231–234, 1997

1997
[26]

On the input- output stability of nonlinear systems: Large gain theorem

V . Zahedzadeh, H.J. Marquez, and T. Chen, “On the input- output stability of nonlinear systems: Large gain theorem”, in Proc. 2008 American Control Conference , pp. 3440 - 3445, 2008

2008
[27]

Freeman, ‘On the Role of Well-Posedness in Homotop y Methods for the Stability Analysis of non-linear Feedback Systems’ , In Lecture Notes in Control and Information Sciences, pp

R.A. Freeman, ‘On the Role of Well-Posedness in Homotop y Methods for the Stability Analysis of non-linear Feedback Systems’ , In Lecture Notes in Control and Information Sciences, pp. 43-82, 2022

2022

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Skogestad and I

S. Skogestad and I. Postletwaite, Multivariable Feedback Control: Analysis and Design . John Wiley & Sons, 2005

work page 2005

[2] [2]

K. J. Astrom and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers . Princeton, NJ: Princeton University Press, 2010

work page 2010

[3] [3]

Scaled relative graphs: Non- expansive operators via 2D Euclidean geometry,

E. K. Ryu, R. Hannah, and W. Yin, “Scaled relative graphs: Non- expansive operators via 2D Euclidean geometry,” Mathematical Pro- gramming, vol. 194, pp. 569–619, 2022

work page 2022

[4] [4]

Scaled relative g raphs for system analysis,

T. Chaffey, F. Forni and R. Sepulchre, “Scaled relative g raphs for system analysis,” in Proc. 60th IEEE Conference on Decision and Control, Austin, TX, USA, pp. 3166-3172, 2021

work page 2021

[5] [5]

A rolled-off passivity theorem,

T. Chaffey, “A rolled-off passivity theorem,” Systems & Control Letters, vol. 162, 2022

work page 2022

[6] [6]

Graphical Nonli near System Analysis,

T. Chaffey, F. Forni, and R. Sepulchre, “Graphical Nonli near System Analysis,” IEEE Transactions on Automatic Control , vol. 68, no. 10, pp. 6067-6081, 2023

work page 2023

[7] [7]

Scaled graphs for reset control system analysis,

S.J.A.M. van den Eijnden, T. Chaffey, T. Oomen, and W.P .M .H. Heemels, “Scaled graphs for reset control system analysis, ” European Journal of Control , vol. 80, art. 101050, 2024

work page 2024

[8] [8]

The scaled relative graph of a linear operator

R. Pates, “The scaled relative graph of a linear operator ”, 2021, arXiv:2106.05650

arXiv 2021