{"id":"4f32d99a-9529-475b-a422-209d8efdfcfe","arxiv_id":"2411.18318","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Nyquist-aware extended Scaled Relative Graph lets SRG analysis handle unstable plants and yields a generalized circle criterion with L2-gain bounds.","lead":"This paper upgrades a graphical tool called Scaled Relative Graphs so that it can analyze feedback systems with unstable parts, and uses the upgrade to prove a broader version of the classic circle criterion. The result provides stability checks and input-output gain bounds for nonlinear systems that are not limited to the usual sector-bounded class.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5's proof applies Proposition 1.c to strictly proper G, whose inverse is not a causal L2e operator; the set inversion of SRG'(G) is asserted without proof.","rationale":"The paper's central claim is that the Nyquist winding information can be merged into the SRG to yield a generalized circle criterion and gain bounds for Lur'e systems with unstable LTI plants. The main theorem is plausible, and the worked example is suggestive, but the proof as written has a genuine gap: it invokes Proposition 1.c for the inverse of a strictly proper LTI operator, even though that inverse is improper and may not be a well-defined causal operator on L2e. This is not a disagreement with consensus; it is an internal-correctness issue about whether Eq. (11), defined by set inversion of SRG'(G), actually implies the hypotheses of Theorem 4. The reader's weakest-assumption analysis identified the same issue, and my independent reading agrees. I also considered whether the step '-1/κ ∉ SRG'(G) implies stability' is flawed because N_R(z)+np could be negative; that concern dissolves because the winding number plus np equals the number of unstable closed-loop poles, which is always nonnegative on the complement of the Nyquist curve. The proposed concrete test settles whether the formal inversion step is valid for the paper's own example; if it passes, the proof can likely be repaired by a direct Nyquist-set argument, so the verdict remains conditional rather than reject.","tokens_in":9841,"tokens_out":12846,"duration_ms":119264,"concrete_test":"For the Section VI example (G(s)=3/((s-2)(s/10+1)), κ=1.5), compute SRG(˜G) directly from Theorem 3 using the stable transfer function ˜G = G/(1+κG), then compute its pointwise inverse (SRG(˜G))^{-1}. Compare this set with (SRG'(G))^{-1} + κ obtained by formal inversion of the extended SRG. If the two sets differ by more than numerical tolerance, or if the distance from (SRG(˜G))^{-1} to -SRG(φ-κ) is not at least 1/4, the proof's key equality fails and Theorem 5 needs an additional argument or a modified hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 5 hinges on the chain SRG(˜G^{-1}) = SRG(G^{-1}) + κ and on Eq. (11) supplying the distance condition required by Theorem 4. For a strictly proper G, which is the class used in the circle criterion and in the Section VI example, G^{-1} is an improper transfer function (e.g., G(s)=1/(s+1) gives G^{-1}=s+1), so it is not a causal operator L2e→L2e and is not defined on all of L2. Proposition 1.c is stated for operators R with a genuine inverse R^{-1} on the Hilbert space L; using it to equate (SRG'(G))^{-1} with the SRG of this nonexistent operator goes outside the stated hypothesis. The extended SRG is defined purely via Nyquist data (Definition 1), so inversion of SRG'(G) is a new set-level operation; no proof shows that it yields the SRG of the post-loop-transformation inverse or that it satisfies Theorem 4's distance condition. Since Eq. (11) and the bound Γ(T) ≤ 1/r are formulated in terms of this inverted set, the main theorem does not follow from the cited propositions as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10030,"tokens_out":11555,"duration_ms":95866,"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":[{"comment":"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.","section":"V-A, proof of Theorem 5"},{"comment":"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.","section":"Definition 1, Eq. (8)"},{"comment":"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.","section":"VI, proof of Theorem 6"},{"comment":"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.","section":"V-A, Theorem 5 statement"}],"minor_comments":[{"comment":"The phrase \"a pitfall that limit its applicability\" should be \"a pitfall that limits its applicability\".","section":"Abstract"},{"comment":"There is a typo in \"Propostion 1.c.\" — it should be \"Proposition 1.c.\".","section":"VI"},{"comment":"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.","section":"Definition 1"},{"comment":"The figures are informative, but the shading and line conventions could be explained more explicitly in the captions to improve readability.","section":"Figures 4 and 5"}],"recommendation":"major_revision","confidential_remarks":"The core idea is promising and the paper is clearly written, but the proof of Theorem 5 currently depends on an unproven extension of SRG calculus to improper operators. I recommend requesting a major revision in which the authors either define the SRG for improper LTI transfer functions via a limiting argument and prove Proposition 1.c for that class, or restate the main theorem directly as a statement about the extended set SRG′(G) and prove the separation condition without invoking the inverse operator. The equivalence in Theorem 6 should be checked carefully; it appears to be false in general. The paper is within the scope of the journal and, if the technical gaps are fixed, would be a valuable contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere'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.\n\nWhere 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.\n\nNow 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.\n\nOverall: 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.\n\nRecommendation: 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.","headline":"Clever fix for a real SRG pitfall, but Theorem 5's proof needs a proper treatment of inverses of strictly proper plants.","tokens_in":10572,"tokens_out":4009,"would_cite":false,"duration_ms":37644,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93D10","93C10","93B52"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Scaled Relative Graph","Lur'e systems","generalized circle criterion","Nyquist criterion","incremental L2-gain","nonlinear feedback stability","well-posedness"],"falsifier":"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.","tokens_in":9580,"feed_emoji":"🔄","tokens_out":6319,"duration_ms":50613,"temperature":0.7,"pith_summary":"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.","feed_headline":"Nyquist info fixes a false-stability flaw in Scaled Relative Graphs","feed_subtitle":"Adding winding-number data to the SRG yields Lur'e-system stability with L2-gain bounds beyond sector nonlinearities.","key_machinery":"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 $1/r$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the Nyquist stability criterion whose encirclement count the extended SRG incorporates.","marker":"[1]"},{"why":"Defines Scaled Relative Graphs and the calculus rules (inversion, chord property, sum and product bounds) used throughout the paper.","marker":"[5]"},{"why":"Introduced SRG analysis of feedback systems, including Theorem 3 on the SRG of stable LTI operators and the stability theorem that Theorem 5 extends.","marker":"[6]"},{"why":"Supplies the homotopy construction that guarantees well-posedness of the feedback interconnection.","marker":"[9]"},{"why":"Provides the updated homotopy theorem for incremental stability on which Theorem 4's well-posedness and gain bound rest.","marker":"[10]"},{"why":"Contains the classical circle criterion (Thm 5.2.10) that the generalized circle criterion extends and is compared against.","marker":"[11]"},{"why":"Prior circle criterion for sector-bounded dynamic nonlinearities, used as a comparison point for the generalization.","marker":"[13]"}],"fun_headline_variants":["SRG gets Nyquist fix for Lur'e systems and L2-gain bounds","Nyquist winding repairs SRG for broader circle criterion","From sector limits to L2-gain: SRG plus Nyquist generalizes","Circle criterion beyond sectors via Nyquist-augmented SRG"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["SRG gets Nyquist fix for Lur'e systems and L2-gain bounds","Nyquist winding repairs SRG for broader circle criterion","From sector limits to L2-gain: SRG plus Nyquist generalizes","Circle criterion beyond sectors via Nyquist-augmented SRG"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00069,"raw_usage":{"total_tokens":3136,"prompt_tokens":970,"completion_tokens":2166,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":586,"completion_tokens_details":{"reasoning_tokens":2089}},"tokens_in":586,"tokens_out":2166,"duration_ms":14069,"temperature":1.0,"reasoning_tokens":2089,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:20:44.647976+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Scaled relative graphs: Non- expansive operators via 2D Euclidean geometry,","cited_arxiv_id":null,"evidence_quote":"Defines Scaled Relative Graphs and the calculus rules (inversion, chord property, sum and product bounds) used throughout the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains the classical circle criterion (Thm 5.2.10) that the generalized circle criterion extends and is compared against."},{"cited_title":"Van Der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control, ser","cited_arxiv_id":null,"evidence_quote":"Prior circle criterion for sector-bounded dynamic nonlinearities, used as a comparison point for the generalization."}],"review_version":1}