{"id":"3c8e36c1-5e50-4e9a-873c-84a1732a1ae7","arxiv_id":"2507.15564","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper adds Nyquist encirclement information to the Scaled Relative Graph (SRG) method so it can handle feedback systems with unstable linear plants. The result is a generalized circle criterion and an L2-gain bound for general nonlinear interconnections.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6's central step — that a bounded computed SRG bound during inflation implies full-domain L2 stability — is asserted without proof; bounded SRGs over dom(R) do not imply dom(R)=L2, the paper's own pitfall.","rationale":"The reader rejects with high confidence, and I agree that the central claim is not established. However, I locate the decisive weakness slightly differently. The reader's weakest assumption — the unproved Section III-A3 equality for unstable LTI SRGs — is real and should be fixed, but it is an ingredient: even if that equality holds, Theorem 6 could still fail. The proof of Theorem 6 makes an inference that the paper itself identifies as the pitfall: a finite radius of a computed SRG bound only controls behavior on dom(R_tau), and the assumptions of continuity and inflatability do not, by themselves, force dom(R_tau)=L2. A homotopy theorem needs either a fixed-point argument or a reduction to a feedback loop with uniform separation for all tau; the OCFG framework provides neither. Consequently the headline general-interconnection theorem is unsupported, and the Section VI examples do not compensate because they either use Theorem 4 (canonical forms with explicit loop transformations) or, in Section VI-C, rely on Theorem 6 itself. Credit where due: the extended-SRG interconnection rules (Theorem 3) and the generalized circle criterion for Lur'e systems are plausible and appear to have independent value; the failure is in the main general claim, not in all components. A major revision should (i) prove or replace Section III-A3, (ii) complete the homotopy proof for general words, or (iii) downgrade Theorem 6 to a claim about gain bounds on dom(R) plus a clearly stated well-posedness assumption.","tokens_in":26795,"tokens_out":8102,"duration_ms":96602,"concrete_test":"Independently re-derive Theorem 6 from the homotopy theorem used for the canonical cases: exhibit, for an arbitrary word wR, a loop transformation that rewrites R_tau in the feedback form (H1^{-1} + H2_tau)^{-1} (or a finite cascade of such forms), with H1 a fixed stable LTI operator and H2_tau having SRG/SG0 bounded by a uniform constant and a uniform separation margin rm>0 for all tau in [0,1]. Apply this to the Section VI-C word with G(s)=3/((s-2)(s/10+1)), K(s)=1/(s+1), and the given saturations. If no such reduction can be produced, the proof step 'We can conclude dom(R_tau)=L2' is unsupported, and Theorem 6 is at best conditional on a missing well-posedness argument.","verdict_should_be":"REJECT","load_bearing_attack":"The load-bearing gap is in the proof of Theorem 6 (Appendix B). After defining the computed bound C(R_tau), the proof states: 'From (20) and the condition that C(R) has finite radius r, we know that R_tau is stable for all tau in [0,1]. We can conclude that dom(R_tau) = L2 for all tau in [0,1].' This is exactly the inference the paper warns against in Section III-A2: a finite radius of an SRG-type set only bounds the (incremental) gain on dom(R_tau); it does not by itself show that every L2 input produces an L2 output. The homotopy continuity and inflatability assumptions do not close this gap unless they are used inside a fixed-point or well-posedness argument (as in Theorem 2 or [10, Cor. 1]) that propagates full-domain solvability from tau=0. No such argument is given for a general word wR; the OCFG construction computes set inclusions, not a family of feedback loops with a uniform separation margin. Thus the central claim that Theorem 6 assesses stability and L2 gain for general interconnections is not established. A second unproved ingredient is the 'clear' equality in Section III-A3 relating SRG_U(T) to the Nyquist hull for unstable LTI T; resolving it alone would not repair Theorem 6.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper identifies a pitfall in existing Scaled Relative Graph (SRG) analysis: for an unstable LTI operator, a finite-radius SRG computed over its domain does not imply L2 stability. To resolve this, the authors propose an extended SRG for LTI operators that adds Nyquist encirclement information to the h-convex hull of the Nyquist diagram. They prove that the extended SRG obeys the same interconnection rules as the ordinary SRG, use it to derive stability and L2-gain conditions for three canonical feedback interconnections (Lur'e, controlled Lur'e, and Lur'e-controlled systems), and obtain a generalized circle criterion. For general interconnections, they introduce an Operator Context-Free Grammar (OCFG) representation and state Theorem 6, which claims that if the linearized interconnection is stable and a bound computed by SRG calculus has finite radius, then the nonlinear interconnection is L2-stable with the corresponding gain bound. Three examples (Duffing oscillator, nonlinear pendulum, and a controlled Lur'e plant with saturation) illustrate the proposed analysis.","tokens_in":27165,"tokens_out":13383,"duration_ms":152975,"significance":"If correct, the paper would make a substantial contribution to graphical nonlinear system analysis: it would extend exact SRG-based stability and performance analysis to interconnections containing unstable LTI plants, provide a rigorous generalization of the circle criterion with L2-gain bounds, and supply a formal-language framework for systematically computing SRG bounds on arbitrary SISO interconnections. The paper is careful in setting up causal operators on L2/L2e, gives proofs for the extended-SRG interconnection rules, and includes worked examples with simulations. However, the central theorem for general interconnections contains a load-bearing logical gap, and a foundational equality for the extended SRG is asserted without proof. These issues prevent acceptance in the current form.","major_comments":[{"comment":"The proof of Theorem 6 states: 'From (20) and the condition that C(R) has finite radius r, we know that R_tau is stable for all tau in [0,1]. We can conclude that dom(R_tau) = L2 for all tau in [0,1].' This inference is exactly the pitfall the paper itself identifies in Section III-A2: a finite radius of an SRG-type set only bounds the (incremental) gain on the domain of the operator, and does not imply that the domain is all of L2. The inclusions in (20) are set inclusions for SRG-type bounds; the continuity and inflatability assumptions give monotone inclusions C(R_tau1) subset C(R_tau2), but no uniform separation margin or fixed-point argument analogous to Theorem 2 or Theorem 10 is supplied for a general word wR in the OCFG. Therefore the conclusion dom(R) = L2e in Theorem 6 is unsupported. A repair would require either a genuine homotopy/well-posedness argument that propagates full-domain solvability from tau=0, or a weakening of the theorem to a gain bound on dom(R).","section":"Section V-B2, Step 4 and Appendix B, proof of Theorem 6"},{"comment":"The paper states that for an unstable LTI operator T, the SRG over the set U of inputs for which Tu is in L2 is 'clear[ly]' equal to the h-convex hull of the Nyquist diagram of T. No proof is provided. This equality is the basis of Definition 3 of the extended SRG and therefore underpins all subsequent extended-SRG results. If the equality fails, the encirclement region NR may not correctly characterize the restricted-domain behavior of unstable LTI operators. This is a load-bearing unproved assertion and should be either proved or stated explicitly as an assumption.","section":"Section III-A3"},{"comment":"The statement of Theorem 6 does not assume causality of R, yet the proof concludes: 'Since R is causal by assumption, we know by Lemma 1 that R : L2e to L2e'. The causality hypothesis is missing from the theorem statement. Without it, Lemma 1 cannot be invoked to extend a domain conclusion from L2 to L2e. The theorem statement should include the causality assumption, or the conclusion should be restricted to dom(R) = L2.","section":"Theorem 6 statement versus proof"}],"minor_comments":[{"comment":"In the definition of continuity in tau, the phrase 'the map u 7-> ||PT R_tau u||2 is continuous in tau' should read 'the map tau 7-> ||PT R_tau u||2 is continuous in tau'; the variable in the map is tau, not u.","section":"Definition 6"},{"comment":"The theorem statement says 'the following statements hold for the SRG defined in Theorem 4', but the extended SRG is defined in Definition 3, not Theorem 4. This cross-reference should be corrected.","section":"Theorem 3"},{"comment":"The derivation of SRG'(K)^{-1} = D[0,1/kp] refers to 'Theorem 7' when invoking the Nyquist criterion; it would be clearer to refer to Definition 3 and Lemma 3, since the statement concerns the extended SRG of an LTI operator.","section":"Section VI-B"},{"comment":"The phrase 'a1 = 0 or a2 = 0 would amount to a pole-zero cancellation' needs a short justification; if a_i(s_u)=0 with b_i(s_u) != 0, then the relevant instability condition fails, so the exclusion is valid but not immediate as written.","section":"Proof of Theorem 3, items 4 and 5"},{"comment":"The informal description says that if RLTI is stable and C(R) has finite radius, then 'the stability of RLTI is not lost during the inflation' and R is stable. This is a restatement of the unsupported inference discussed in the first major comment; it should be revised to reflect the actual assumptions needed.","section":"Section V-B2, informal Step 4"}],"recommendation":"reject","confidential_remarks":"The main obstacle is not presentation but a load-bearing gap in the central theorem: Theorem 6's proof repeats the very pitfall the paper identifies. The missing causality assumption in the theorem statement further weakens the L2e conclusion. The Section III-A3 equality, while plausible, is asserted without proof and is foundational to the extended SRG. I would be willing to consider a revised version that supplies a valid homotopy or small-gain argument for general OCFG interconnections and a proof of the restricted-domain SRG equality; as it stands, the advertised claim of a stability and L2-gain theorem for general interconnections is not established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things. The extended SRG and its calculus (Definition 3, Theorem 3) are a real step forward for the SRG literature. The paper also correctly identifies a genuine pitfall: a bounded SRG computed over dom(R) does not imply dom(R)=L2. The examples are carefully done, and the generalized circle criterion is a useful extension.\n\nThe weak spot is Theorem 6. In the proof in Appendix B, the step from a finite radius of C(R) to 'R_tau is stable for all tau' to 'dom(R_tau)=L2' is exactly the inference the paper itself warns against in Section III-A2. A finite radius of an SRG-type set only bounds the gain on dom(R_tau); it doesn't make the domain all of L2. The continuity and inflatability assumptions do not close this gap unless they are used in a fixed-point or well-posedness argument that propagates full-domain solvability from tau=0 (as in Theorem 2 via [10, Cor. 1]). The OCFG construction computes set inclusions, not a family of feedback loops with a uniform separation margin. So the headline claim—a stability and L2-gain theorem for general interconnections—is not established as written.\n\nA second, smaller gap: Section III-A3 says it is 'clear' that for an unstable LTI operator, the SRG over the stabilizing input set is the h-convex hull of the Nyquist diagram. This is a restricted-domain statement that needs proof; if it is false, Definition 3's encirclement region may not describe the restricted operator. I suspect it is true, but it should not be asserted.\n\nWhat survives? Theorem 4 for the Lur'e-type canonical interconnections looks largely sound, because well-posedness there is supplied by Theorem 2 rather than by the broken step in Theorem 6. The examples are worked honestly and the simulations match the gain bounds. The formal-language approach is more bookkeeping than deep mathematics, but it is a reasonable way to organize computations.\n\nThis paper deserves a serious referee. The core ideas are good, but Theorem 6 needs either a repaired proof—likely by showing a uniform separation margin along the homotopy and invoking a fixed-point theorem—or a weakened statement that only claims gain bounds on dom(R). I would not desk-reject; I would ask for major revision on Theorem 6 and the III-A3 claim.","headline":"Extended SRG is a real advance, but the general-interconnection theorem rests on the exact domain inference the paper warns against.","tokens_in":27606,"tokens_out":8640,"would_cite":true,"duration_ms":93886,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C10","93C80","93D10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Adding Nyquist encirclement information to the scaled relative graph lets one certify stability and $L_2$-gain for feedback systems with unstable plants and general nonlinearities, yielding a generalized circle criterion.","keywords":["Scaled Relative Graph","nonlinear stability","L2-gain","Nyquist criterion","circle criterion","Lur'e systems","feedback interconnections","context-free grammar"],"falsifier":"Compute $SRG_U(T)$ for an unstable LTI operator $T$ on a truncated finite-dimensional approximation of the stabilizing input set $U$ and compare it with the h-convex hull of $Nyquist(T)$; a single pair of finite-energy inputs $u_1,u_2$ whose ratio $z_T(u_1,u_2)$ lies outside the hull would falsify the equality that Definition 3 relies on. A sharper test is to check Theorem 3.c: find a candidate $G$ with $n_p > 0$ for which the inversion rule fails by comparing $N_G$ and $N_{G^{-1}}$ on a point not on the real axis, since the proof only establishes the real-axis correspondence.","tokens_in":26598,"feed_emoji":"📈","tokens_out":9352,"duration_ms":89329,"temperature":0.7,"pith_summary":"The paper identifies a pitfall in scaled relative graph (SRG) analysis of nonlinear feedback systems: for an unstable linear time-invariant plant inside a loop, ordinary SRG calculus can produce a finite gain bound for the closed loop even when the Nyquist criterion says the loop is unstable. The reason is that the SRG is defined only on the restricted input domain for which the unstable operator maps finite-energy signals to finite-energy signals, so a finite graph radius does not certify stability on all of $L_2$. The paper's fix is an extended SRG that adds the Nyquist encirclement region to the h-convex hull of the Nyquist diagram, and it proves that this extended object obeys the same interconnection rules as the ordinary SRG. With that, the paper derives a generalized circle criterion for Lur'e systems and a stability and incremental/non-incremental $L_2$-gain theorem for arbitrary interconnections of SISO LTI and nonlinear operators. This matters because it extends an exact graphical method from stable open-loop plants to the unstable plants and general nonlinearities that appear in practical feedback design.","feed_headline":"Nyquist fix lets SRG analysis handle unstable nonlinear plants","feed_subtitle":"Adding encirclement data to scaled relative graphs yields a generalized circle criterion with L2-gain bounds.","key_machinery":"The extended SRG: for an LTI operator $R$, the h-convex hull $G_R$ of its Nyquist diagram is merged with the encirclement region $N_R = \\{z : N_R(z)+n_p > 0\\}$, where $N_R(z)$ is the clockwise winding number of the Nyquist curve around $z$ and $n_p$ is the number of right-half-plane poles. The extra region is what carries the Nyquist criterion's stability information that the ordinary SRG discards. The argument also rests on two supporting mechanisms: a homotopy 'inflation' of each nonlinear operator from a real gain $\\kappa \\in SRG(\\phi)$ to the full nonlinearity, which lets stability of the linearized loop imply stability of the nonlinear loop; and a context-free grammar representation of interconnections, which turns block diagrams into words whose SRG bounds can be computed by the algebraic rules.","core_discovery":"On its own terms, the paper's central claim is that the object $SRG'(R) = G_R \\cup N_R$, where $G_R$ is the h-convex hull of the Nyquist diagram of an LTI operator $R$ and $N_R = \\{z : N_R(z)+n_p > 0\\}$ is the set of points whose clockwise winding number plus the number of unstable poles is positive, behaves exactly like the ordinary SRG under the algebraic operations of scaling, adding the identity, inversion, summation, and product (Theorem 3). This extended SRG resolves the apparent contradiction between SRG calculus and the Nyquist criterion. Using it, the paper proves that a Lur'e feedback loop is stable with $L_2$-gain bound $1/r_m$ whenever $\\operatorname{dist}(SRG'(G)^{-1}, -SRG(\\phi)) \\ge r_m > 0$, for any inflatable nonlinear operator $\\phi$, not only sector-bounded nonlinearities (Theorems 4 and 5). For general interconnections, the paper represents the interconnection as a word in a context-free grammar over LTI and nonlinear operator symbols, replaces each symbol by its (extended) SRG, and proves that the resulting set $C(R)$ satisfies $\\Gamma(R) \\le r_{\\min}(C(R))$ on $\\operatorname{dom}(R)$, with $\\operatorname{dom}(R) = L_{2e}$ under continuity and inflation assumptions (Theorem 6).","pith_inferences":["The paper's key unproved equality — that the SRG over finite-energy stabilizing inputs equals the h-convex hull of the Nyquist diagram for unstable LTI operators — is the load-bearing point; if it only holds for restricted classes of transfer functions, the extended SRG's definition would need modification, and testing this equality on non-minimum-phase or repeated-pole examples is a natural next ","The same 'fill the hole with encirclement data' repair could be applied to other restricted-domain graphical tools, such as nonlinear Bode diagrams or harmonic-balance methods, whenever the underlying frequency response is defined only on stabilizing inputs.","Because different grammar words for the same operator yield different sets $C(R)$, the framework opens an algorithmic problem: search over equivalent interconnections and over $\\kappa$ to minimize the radius of the bound, analogous to multiplier optimization.","The generalized circle criterion suggests that any bounded, inflatable operator with a known SRG — including dynamic or reset nonlinearities — can be plugged into classical Lur'e-type stability arguments, which may extend $L_2$-gain performance shaping to control architectures where sector conditions fail."],"forward_implications":["For Lur'e systems, the classical circle criterion is recovered as a special case, and the new criterion applies to time-varying sector nonlinearities, reset elements, and other operators whose SRG is bounded and inflatable, while also supplying an $L_2$-gain bound.","Feedback loops containing unstable plants or integrators can be analyzed by direct SRG calculus without the false finite-gain conclusions that occur with the ordinary SRG.","Arbitrary finite interconnections of SISO LTI and nonlinear operators can be certified stable with incremental or non-incremental $L_2$-gain bounds by computing a single set $C(R)$ from the interconnection's grammar word.","The choice of the real gain $\\kappa$ in the inflation acts as a loop transformation or multiplier that can be tuned to maximize the separation distance $r_m$ and thus tighten the gain bound.","Well-posedness and causality of the closed-loop system follow from the same separation conditions whenever the subsystems are causal."],"supporting_citations":[{"why":"Introduces the scaled relative graph and the connection rules for sums, products, inverses, and scaling that Theorem 3 extends to the extended SRG.","marker":"[5]"},{"why":"Establishes that the SRG of a stable LTI operator is the h-convex hull of its Nyquist diagram and gives the simple-feedback stability theorem that the paper improves.","marker":"[6]"},{"why":"Supplies the homotopy theorem for incremental stability used to justify inflating nonlinearities and proving well-posedness.","marker":"[10]"},{"why":"The conference-paper predecessor that proved the Lur'e and generalized circle criterion results which this paper generalizes to arbitrary interconnections.","marker":"[9]"},{"why":"Provides the input-output feedback stability and well-posedness definitions underlying the small-gain and homotopy arguments.","marker":"[12]"},{"why":"Provides the causality and extended-space $L_{2e}$ background used to extend gain bounds from $L_2$ to $L_{2e}$.","marker":"[13]"},{"why":"Supplies the context-free grammar machinery used to represent general interconnections as words.","marker":"[18]"},{"why":"Used alongside the homotopy construction to guarantee well-posedness when inflating nonlinearities.","marker":"[11]"}],"fun_headline_variants":["SRG plus Nyquist yields generalized circle criterion with gain bounds","Fixed SRG now handles unstable nonlinear plants via encirclement","Encirclement data extends scaled relative graphs to unstable systems","Generalized circle criterion from SRG and Nyquist for nonlinear loops","New SRG tool gives stability and L2-gain for nonlinear interconnections"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that for an unstable linear time-invariant operator, the scaled relative graph computed over exactly the finite-energy inputs that the operator keeps finite-energy equals the h-convex hull of its Nyquist diagram; the paper states this as clear rather than proving it, and Definition 3 of the extended SRG is built on it.","fun_headline_variants_meta":{"raw":{"variants":["SRG plus Nyquist yields generalized circle criterion with gain bounds","Fixed SRG now handles unstable nonlinear plants via encirclement","Encirclement data extends scaled relative graphs to unstable systems","Generalized circle criterion from SRG and Nyquist for nonlinear loops","New SRG tool gives stability and L2-gain for nonlinear interconnections"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000827,"raw_usage":{"total_tokens":3640,"prompt_tokens":998,"completion_tokens":2642,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":614,"completion_tokens_details":{"reasoning_tokens":2553}},"tokens_in":614,"tokens_out":2642,"duration_ms":17508,"temperature":1.0,"reasoning_tokens":2553,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:30:57.076104+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $SRG_U(T)$ for an unstable LTI operator $T$ on a truncated finite-dimensional approximation of the stabilizing input set $U$ and compare it with the h-convex hull of $Nyquist(T)$; a single pair of finite-energy inputs $u_1,u_2$ whose ratio $z_T(u_1,u_2)$ lies outside the hull would falsify the equality that Definition 3 relies on. A sharper test is to check Theorem 3.c: find a candidate $G$ with $n_p > 0$ for which the inversion rule fails by comparing $N_G$ and $N_{G^{-1}}$ on a point not on the real axis, since the proof only establishes the real-axis correspondence.","supporting_citations":[{"cited_title":"Hopcroft, Rajeev Motwani, and Jeffrey D","cited_arxiv_id":null,"evidence_quote":"Supplies the context-free grammar machinery used to represent general interconnections as words."}],"review_version":1}