Scaled Relative Graph Analysis of Lur'e Systems and the Generalized Circle Criterion

Amritam Das; Julius P.J. Krebbekx; Roland T\'oth

REVIEW 4 major objections 4 minor 2 cited by

Scaled Relative Graph Analysis of Lur'e Systems and the Generalized Circle Criterion

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that merging Nyquist encirclement information into the Scaled Relative Graph removes a stability blind spot and yields a generalized circle criterion for Lur'e systems with arbitrary nonlinear operators.

desk verdict Clever fix for a real SRG pitfall, but Theorem 5's proof needs a proper treatment of inverses of strictly proper plants. read the letter →

arxiv 2411.18318 v2 pith:QJWCVLZ4 submitted 2024-11-27 eess.SY cs.SYmath.OC

Julius P.J. Krebbekx , Roland Tóth , Amritam Das This is my paper · ORCID

classification eess.SYcs.SYmath.OC MSC 93D1093C1093B52

keywords ScaledRelativeGraphLur'esystemsgeneralizedcirclecriterionNyquistincrementalL2-gainnonlinearfeedbackstabilitywell-posedness

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 claims that Scaled Relative Graphs (SRGs), a graphical method that represents a nonlinear operator as a set of complex numbers, can give false stability verdicts when the LTI plant in a feedback loop is unstable, because the SRG discards the frequency and encirclement information that the Nyquist criterion uses. To fix this, the paper defines an extended SRG for LTI operators that adds the winding-number region to the usual h-convex hull of the Nyquist diagram. Using this extended SRG in a loop-transformation argument, it proves a stability and well-posedness theorem for Lur'e systems with a gain bound of $\Gamma(T) \le 1/r$, and derives a generalized circle criterion that applies to general nonlinear operators, not just sector-bounded ones. A sympathetic reader would care because this extends an exact, graphical, performance-quantifying method to the practically important case of stabilizing unstable plants.

What carries the argument

The central object is the extended Scaled Relative Graph $\mathrm{SRG}'(R) := \mathcal{G}_R \cup \mathcal{N}_R$, where $\mathcal{G}_R$ is the h-convex hull of the Nyquist diagram of an LTI operator $R$ and $\mathcal{N}_R$ is the set of points $z$ with $\mathcal{N}_R(z) + n_p > 0$, with $n_p$ the number of open right-half-plane poles. This object carries the argument because it injects the Nyquist encirclement count, previously absent from SRG analysis, into the graphical set, so that the distance separation in Eq. (11) reflects both gain and stability information. The proof also relies on a loop transformation with $\kappa \in \mathrm{SRG}(\phi)$, which shifts the nonlinearity's SRG to make the transformed plant stable and lets the prior stability theorem (Theorem 4) apply with a gain bound $1/r$.

What would settle it

Compute the actual closed-loop incremental gain of a strictly proper unstable plant and a nonlinearity satisfying Eq. (10) for which Eq. (11) holds with a stated $r$; if $\Gamma(T) > 1/r$ or the loop is unstable, the inversion step is invalid. More directly, evaluate $\mathrm{SRG}(G^{-1})$ and $\mathrm{SRG}(G)^{-1}$ for a strictly proper $G$ with a right-half-plane pole over finite-dimensional truncations of $L_{2e}$ and check whether the sets coincide.

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Extended reading notes

Core claim

The central discovery is that the apparent contradiction between SRG calculus and Nyquist theory arises because the SRG of an unstable LTI operator, taken over the set of stabilizing signals, is the h-convex hull of its Nyquist diagram, which erases the winding number. The fix is the extended SRG $\mathrm{SRG}'(G) = \mathcal{G}_G \cup \mathcal{N}_G$, where $\mathcal{N}_G$ collects the points whose clockwise encirclement count plus the number of unstable poles is positive. The main theorem (Theorem 5) states that if one of $\mathrm{SRG}(G)^{-1}$ or $\mathrm{SRG}(\phi)$ obeys the chord property and some real $\kappa \in \mathrm{SRG}(\phi)$ makes Eq. (10) hold, then the distance condition $\mathrm{dist}(\mathrm{SRG}'(G)^{-1}, -\mathrm{SRG}(\phi)) \ge r > 0$ implies that the closed-loop operator $T = (G^{-1} + \phi)^{-1}$ is stable, well-posed, and has incremental $L_2$-gain $\Gamma(T) \le 1/r$. From this, the paper derives Theorem 6, a generalized circle criterion: separation of $\mathrm{SRG}'(G)$ from $-\mathrm{SG}_0(\phi)^{-1}$ guarantees $L_2$-boundedness with $\gamma(T) \le 1/r_m$, and replacing $\mathrm{SG}_0$ by $\mathrm{SRG}$ upgrades the bound to incremental gain.

Load-bearing premise

The proof of Theorem 5 assumes that the SRG inversion rule $\mathrm{SRG}(G^{-1}) = (\mathrm{SRG}(G))^{-1}$ remains valid when $G$ is strictly proper, in which case $G^{-1}$ is not a causal operator on $L_{2e}$; the paper does not justify this non-causal inversion, and the entire loop-transformation and distance argument depends on it.

Editorial extensions

If this is right

The SRG method can now be used to analyze feedback loops with unstable LTI plants, which the previous SRG stability theorem explicitly excluded.
The generalized circle criterion in Theorem 6 extends stability guarantees to nonlinear operators beyond incremental sector bounds, provided the SRG distance condition and chord and star-shaped conditions hold.
Unlike the classical circle criterion, which only certifies boundedness, the extended-SRG result quantifies (incremental) $L_2$-gain through the inverse of the separation distance $r$.
Well-posedness of the closed-loop operator is guaranteed in the incremental case via the homotopy construction, so the result covers existence, uniqueness, and continuity of the feedback map.
The extended SRG remains at least as sharp as the circle criterion for sector-bounded nonlinearities, since the h-convex hull does not introduce extra conservatism as argued in Remark 4.

Reading between the lines

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

If the extended SRG works as claimed, one could build graphical loop-shaping tools for nonlinear systems: shaping $\mathrm{SRG}'(G)$ to satisfy the separation condition would be the nonlinear analogue of shaping a Nyquist plot, and the $1/r$ bound would give a direct performance readout.
The theorem's star-shaped condition Eq. (10) could be relaxed: Remark 1 already replaces it by checking the distance for all $\tau \in [0,1]$, and a natural next step is to characterize the largest class of nonlinearities that satisfy this condition.
The proof's reliance on inverting strictly proper $G$ is worth probing: if $\mathrm{SRG}(G^{-1}) = \mathrm{SRG}(G)^{-1}$ fails for non-causal inverses, the distance condition would need to be stated in terms of a causal right-inverse or descriptor realization, which could change the graphical condition.
A direct numerical test would be to apply the separation condition to nonlinearities with memory, such as hysteresis or backlash operators, and compare the predicted $1/r$ bound with simulation-based gain estimates to reveal how tight the bound is.

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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,

Here's my read of Krebbekx, Tóth, Das. The paper does two real things: it shows that plain SRG calculus gives false stability predictions for unstable open-loop plants, and it patches this by folding Nyquist winding data into an extended SRG. The generalized circle criterion (Theorem 6) for non-sector nonlinearities with gain bounds is a genuine extension of the classical result, not a repackaging. The example in Section VI is convincing as a proof of concept.

Where it's solid: the pitfall discussion is sharp and the loop-transformation argument is the right high-level idea. The homotopy theorem is used honestly, and the presentation is clear. The claimed new definitions (extended SRG, Theorems 5–6) are not in the cited SRG literature.

Now the soft spots, which are real. The stress-test concern lands: the proof of Theorem 5 applies Proposition 1.c to strictly proper G, whose inverse isn't a causal L2e operator. Proposition 1.c is stated for operators with genuine inverse on the Hilbert space. The set inversion of SRG'(G) is asserted without proof, and the distance condition Eq. (11) is then used as if it were the SRG of the post-transformation inverse. This is a gap in the proof as written, though I suspect the result is true and fixable by a more careful treatment of unbounded inverses (e.g., defining SRG' and its inversion directly from Nyquist data and proving a separation lemma). There are also smaller formal loose ends: Definition 1 doesn't specify N_R for points on the Nyquist curve or for integrators, and the equivalence between Eq. (12) and the distance condition used for the gain bound slides over set-inversion subtleties when ∞ is involved.

Overall: this is a worthwhile paper with a load-bearing proof gap. The central idea is clean and likely correct, but the current write-up doesn't justify the operator-theoretic steps. A serious referee would be justified in asking for a rewrite of the proof of Theorem 5.

Recommendation: yes, send to peer review, but expect heavy revision. The paper is for researchers in nonlinear graphical analysis and control; it deserves a serious referee.

Referee Report

4 major / 4 minor

Summary. The paper identifies a pitfall in Scaled Relative Graph (SRG) analysis of feedback systems: standard SRG calculus applied to a stable LTI loop can predict a finite incremental L2 gain for a closed-loop system that is in fact Nyquist-unstable, because the SRG discards the winding-number (encirclement) information that the Nyquist criterion uses. The authors propose to repair this by defining an extended SRG, SRG′(R), which adds to the h-convex hull of the Nyquist diagram the set of points whose winding number indicates unstable closed-loop behavior (Definition 1). Their main result, Theorem 5, states that a separation condition between the inverted extended SRG of the plant and the SRG of the nonlinearity implies well-posedness and an incremental L2-gain bound for the Lur'e system, via a loop transformation and an application of the homotopy-based Theorem 4. Theorem 6 is presented as a generalized circle criterion for nonlinear operators beyond sector bounds. The paper includes a worked example with an unstable plant and a dead-zone-like nonlinearity, and it argues that the h-convex hull does not introduce conservatism relative to the classical circle criterion.

Significance. If the main result is correct, the paper would make a useful contribution: it addresses a real limitation of current SRG tools, extends graphical frequency-domain stability analysis to unstable LTI plants in feedback with a broad class of nonlinearities, and provides quantitative (incremental) L2-gain bounds with a geometric separation condition. The identification of the pitfall in Section IV and the proposed conceptual remedy are valuable, and the worked example in Section VI is reproducible and visually convincing. The paper does not provide machine-checked proofs or reproducible code, but the derivations are parameter-free and the graphical conditions are concrete. The main concern is that the proof of the central theorem currently relies on an unproven extension of SRG calculus to improper inverse operators and on a questionable set-inversion distance equivalence; these issues must be resolved before the result can be considered established.

major comments (4)

[V-A, proof of Theorem 5] The proof applies the identity SRG(˜G^{-1}) = SRG(G^{-1}) + κ and Proposition 1.c to a strictly proper plant G; for strictly proper G, G^{-1} is an improper transfer function that is not a causal operator on L2e, so SRG(G^{-1}) is not defined under the paper's definition of the SRG for operators on a Hilbert space, and the set-level inversion of SRG′(G) used in Eq. (11) is not justified by the cited propositions. The paper needs either a rigorous definition of the SRG for improper transfer functions (with a proof that Proposition 1.c holds for that class) or a direct proof that the set (SRG′(G))^{-1} satisfies the separation condition required by Theorem 4.
[Definition 1, Eq. (8)] The symbol NR is used both for the set and for the winding-number function, and the definition does not specify how to define NR(z) for z lying on the Nyquist contour or how to handle poles on the imaginary axis in the D-contour; since the proof of Theorem 5 uses the condition −1/κ ∉ SRG′(G) to infer Nyquist stability of ˜G, these ambiguities affect the validity of the main result.
[VI, proof of Theorem 6] The proof of Theorem 6 asserts that Eq. (12), dist(SRG′(G), −SG0(ϕ)^{-1}) > 0, is equivalent to dist(SRG′(G)^{-1}, −SG0(ϕ)) ≥ rm > 0; however, the map z ↦ 1/z does not preserve distances between sets, and no argument is given that these particular h-convex hulls and disks have the required property, so the claimed equivalence is unsupported and appears to be false in general.
[V-A, Theorem 5 statement] Theorem 5 is phrased in terms of SRG(G)^{-1} and SRG(ϕ), but Section IV-C establishes that SRG(G) is not defined for unstable LTI operators, and the example in Section VI applies the theorem to an unstable G using SRG′(G); the theorem statement should be reformulated consistently in terms of the extended SRG.

minor comments (4)

[Abstract] The phrase "a pitfall that limit its applicability" should be "a pitfall that limits its applicability".
[VI] There is a typo in "Propostion 1.c." — it should be "Proposition 1.c.".
[Definition 1] The definition of SRG′(R) does not address poles on the imaginary axis; if the theory is intended to cover integrators (as suggested in Remark 3), the Nyquist D-contour indentation must be specified.
[Figures 4 and 5] The figures are informative, but the shading and line conventions could be explained more explicitly in the captions to improve readability.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main theorem rests on external SRG calculus and homotopy results; the lone self-citation is contextual.

full rationale

The derivation chain of Theorem 5 is built on external results rather than on the paper's own conclusions. The proof applies Proposition 1.c (SRG inversion) and Proposition 1.b (SRG of identity-plus-operator), both attributed to the externally established SRG calculus in [5], and then invokes Theorem 4, which is attributed to [6] and updated in [10]; none of these are authored by the present authors or fitted to the target example. The loop-transformation identity SRG(tilde-G^{-1}) = SRG(G^{-1}) + kappa is an application of the stated SRG algebra, not a restatement of the conclusion. Definition 1 does transparently build Nyquist winding information into SRG'(G) via N_R, and the proof step 'By Eq. (11), -1/kappa notin SRG'(G), hence tilde-G is stable according to the Nyquist criterion' is definitional in the sense that the extended SRG was explicitly constructed to carry Nyquist data. That is an open design choice, not a hidden circularity: the theorem's nontrivial content, namely well-posedness and the incremental L2-gain bound Gamma(T) <= 1/r, is supplied by the external homotopy/SRG separation theorem, and the generalized circle criterion extends the classical one to nonlinearities beyond sector bounds. The only self-citation is [8], a pointer in the introduction to the authors' related work on nonlinear bandwidth and Bode diagrams; it is not used in any proof and is therefore not load-bearing. The reviewer's concern about applying Proposition 1.c to a strictly proper G, whose inverse is not a causal L2e operator, is a mathematical correctness or missing-hypothesis issue, not a circularity: it does not make the theorem's conclusion equivalent to its assumptions by construction. Overall, the paper does not reduce to a fit, a renamed known result, or a self-citation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted numerical constants and no physical entities. Its main construction, the extended SRG SRG', is a formal definition rather than an empirical postulate. The proof depends on the cited homotopy theorem [10] and on algebraic SRG inversion for possibly improper inverse operators.

assumptions (5)

standard math Nyquist stability criterion (Theorem 1) is assumed as background: nz = nn + np.
Used in Definition 1 to define N_R as points where the closed loop would be unstable, and in the proof of Theorem 5 to conclude tilde G is stable.
domain assumption The homotopy theorem (Theorem 4) from Chaffey, Kharitenko, Forni and Sepulchre [10] is assumed as a black box.
The proof of Theorem 5 applies Theorem 4 to the loop-transformed pair (tilde G, tilde phi); the main result inherits its well-posedness and gain-bound conclusions.
ad hoc to paper SRG set calculus rules (Proposition 1) remain valid when applied to inverse operators that are not proper or not causal on L2e.
Theorem 5 uses SRG(G^{-1}) and SRG(tilde G^{-1}) for strictly proper G, where G^{-1} is a differentiator-type non-causal relation; the paper does not justify this extension.
domain assumption For an unstable LTI plant restricted to the stabilizing input set U, SRG_U(L) equals the h-convex hull of Nyquist(L).
Section IV-C states this as 'it is clear' with no proof; it underpins the interpretation of SRG'(L) as the relevant extended object.
domain assumption The Nyquist diagram of a plant with imaginary-axis poles can be extended to a D-contour with well-defined winding numbers.
Remark 3 claims Theorem 5 rigorously handles integrators, but the definition of N_R and SRG' uses the bare frequency response and does not specify contour indentations.

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Cite this review

Pith. "Pith review of Scaled Relative Graph Analysis of Lur'e Systems and the Generalized Circle Criterion." pith.science (2026). https://pith.science/paper/QJWCVLZ4

@misc{pith2026241118318,
  author       = {Pith},
  title        = {Pith review of: Scaled Relative Graph Analysis of Lur'e Systems and the Generalized Circle Criterion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QJWCVLZ4}},
  note         = {Machine review of arXiv:2411.18318}
}

abstract

Scaled Relative Graphs (SRGs) provide a novel graphical frequency-domain method for the analysis of nonlinear systems. However, we show that the current SRG analysis suffers from a pitfall that limit its applicability in analyzing practical nonlinear systems. We overcome this pitfall by modifying the SRG of a linear time invariant operator, combining the SRG with the Nyquist criterion, and apply our result to Lur'e systems. We thereby obtain a generalization of the celebrated circle criterion, which deals with a broader class of nonlinearities, and provides (incremental) $L_2$-gain performance bounds.

Figures

Figures reproduced from arXiv: 2411.18318 by the authors.

**Figure 1.** A simple linear feedback system. When U = L, we denote SRGL(R) = SRG(R). Note that the SRG is a subset of C. One can also define the Scaled Graph (SG) around some particular input. The SG of an operator R with one input u ⋆ ∈ L fixed and the other in set U ⊂ L is defined as SGU,u⋆ (R) := {zR(u, u⋆ ) | u ∈ U}. (5) We introduce the shorthand SGL,u⋆ (R) = SGu⋆ (R). By definition of the SRG (SG), the (non-)incremental g… view at source ↗

**Figure 2.** Block diagram of a Lur’e system. one is interested in the stability and performance aspects of this setup. In this work, we study the stability property. Theorem 1. Let nz denote the number of unstable closedloop poles, and np the number of unstable poles of L(s). Additionally, denote nn the amount of times that L(jω) encircles the point −1 in clockwise fashion as ω traverses the D-contour, going from −jR to jR and… view at source ↗

**Figure 3.** Block diagram of a general feedback interconnection. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: SRGs and Nyquist diagram corresponding to [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: Figures for the SRG analysis of the example in Section VI: (a) graph of the nonlinearity [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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The authors introduce an extended Scaled Relative Graph that includes Nyquist encirclement data, enabling stability and L2-gain analysis of feedback interconnections with unstable linear components.
A Dissipativity Framework for Constructing Scaled Graphs
math.OC 2025-07 conditional novelty 6.0 of 10

A dissipativity-based LMI framework computes scaled graphs for LTI, reset, and piecewise-linear systems, with an exactness guarantee for normal LTI systems.

Reference graph

Works this paper leans on

14 extracted references · 7 canonical work pages · cited by 2 Pith papers

[1]

Regeneration Theory,

H. Nyquist, “Regeneration Theory,” Bell System Technical Journal , vol. 11, no. 1, pp. 126–147, Jan. 1932

1932
[2]

A Frequency-Domain Condition for the Stability of Feedback Systems Containing a Single Time-Varying Nonlinear Element,

I. W. Sandberg, “A Frequency-Domain Condition for the Stability of Feedback Systems Containing a Single Time-Varying Nonlinear Element,” Bell System Technical Journal , vol. 43, no. 4, pp. 1601– 1608, Jul. 1964

work page 1964
[3]

N. M. Krylov, N. N. Bogoliubov, and S. Lefschetz, Introduction to Non-Linear Mechanics , ser. Annals of Mathematics Studies; Annals of Mathematics Studies; Annals of Mathematics Studies; No. 11; No

work page
[4]

Frequency domain performance analysis of nonlinearly controlled motion systems,

A. Pavlov, N. Van De Wouw, A. Pogromsky, M. Heertjes, and H. Nijmeijer, “Frequency domain performance analysis of nonlinearly controlled motion systems,” in Proc. of the 46th IEEE Conference on Decision and Control , 2007, pp. 1621–1627

work page 2007
[5]

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, no. 1-2, pp. 569–619, Jul. 2022

work page 2022
[6]

Graphical Nonlinear System Analysis,

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

2023
[7]

Scaled graphs for reset control system analysis,

S. Van Den Eijnden, T. Chaffey, T. Oomen, and W. (Maurice) Heemels, “Scaled graphs for reset control system analysis,” European Journal of Control, p. 101050, Jun. 2024

work page 2024
[8]

Nonlinear Bandwidth and Bode Diagrams based on Scaled Relative Graphs

J. P. J. Krebbekx, R. T ´oth, and A. Das, “Nonlinear Bandwidth and Bode Diagrams based on Scaled Relative Graphs.” arXiv:2504.01585, 2025

arXiv 2025

Show all 14 references

[9]

System analysis via integral quadratic constraints,

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

1997
[10]

A homotopy theorem for incremental stability,

T. Chaffey, A. Kharitenko, F. Forni, and R. Sepulchre, “A homotopy theorem for incremental stability,” arXiv:2412.01580, 2024

2024 arXiv
[11]

Princeton: Princeton University Press, 1947

1947
[12]

C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, ser. Electrical Science Series. New York: Acad. Press, 1975

1975
[13]

Van Der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control, ser

A. Van Der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control, ser. Communications and Control Engineering. Cham: Springer International Publishing, 2017

2017
[14]

The circle criterion for a class of sector- bounded dynamic nonlinearities,

C. Guiver and H. Logemann, “The circle criterion for a class of sector- bounded dynamic nonlinearities,” Mathematics of Control, Signals, and Systems, vol. 34, no. 3, pp. 461–492, Sep. 2022

2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Regeneration Theory,

H. Nyquist, “Regeneration Theory,” Bell System Technical Journal , vol. 11, no. 1, pp. 126–147, Jan. 1932

1932

[2] [2]

A Frequency-Domain Condition for the Stability of Feedback Systems Containing a Single Time-Varying Nonlinear Element,

I. W. Sandberg, “A Frequency-Domain Condition for the Stability of Feedback Systems Containing a Single Time-Varying Nonlinear Element,” Bell System Technical Journal , vol. 43, no. 4, pp. 1601– 1608, Jul. 1964

work page 1964

[3] [3]

N. M. Krylov, N. N. Bogoliubov, and S. Lefschetz, Introduction to Non-Linear Mechanics , ser. Annals of Mathematics Studies; Annals of Mathematics Studies; Annals of Mathematics Studies; No. 11; No

work page

[4] [4]

Frequency domain performance analysis of nonlinearly controlled motion systems,

A. Pavlov, N. Van De Wouw, A. Pogromsky, M. Heertjes, and H. Nijmeijer, “Frequency domain performance analysis of nonlinearly controlled motion systems,” in Proc. of the 46th IEEE Conference on Decision and Control , 2007, pp. 1621–1627

work page 2007

[5] [5]

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, no. 1-2, pp. 569–619, Jul. 2022

work page 2022

[6] [6]

Graphical Nonlinear System Analysis,

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

2023

[7] [7]

Scaled graphs for reset control system analysis,

S. Van Den Eijnden, T. Chaffey, T. Oomen, and W. (Maurice) Heemels, “Scaled graphs for reset control system analysis,” European Journal of Control, p. 101050, Jun. 2024

work page 2024

[8] [8]

Nonlinear Bandwidth and Bode Diagrams based on Scaled Relative Graphs

J. P. J. Krebbekx, R. T ´oth, and A. Das, “Nonlinear Bandwidth and Bode Diagrams based on Scaled Relative Graphs.” arXiv:2504.01585, 2025

arXiv 2025