{"id":"441566dc-086c-4660-9293-90256940a46c","arxiv_id":"2507.08411","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"A dissipativity-based LMI framework computes scaled graphs for LTI, reset, and piecewise-linear systems, with an exactness guarantee for normal LTI systems.","lead":"This paper gives a new way to compute scaled graphs, a graphical tool for analyzing nonlinear feedback systems, using matrix inequalities and convex optimization. The method is exact for a class of linear systems and gives tighter approximations for reset and piecewise-linear systems than earlier work.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reset-system Theorem 15 depends on lim x(t)=0 via Desoer–Vidyasagar for absolutely continuous functions, but reset trajectories are only right-continuous piecewise absolutely continuous; x, ẋ ∈ L2 does not imply x→0 for signals with jumps, so the soft-IQC step is not justified.","rationale":"The reader's weakest assumption identifies exactly this issue: the extension of the Desoer–Vidyasagar limit result to reset trajectories is not justified. I focused on this rather than on the empty-interior point in Theorem 12 because the exactness theorem's degenerate case can likely be patched by a supporting-half-plane argument or by direct computation, whereas the reset-system limit step is the bridge from the finite-horizon dissipation inequality to the global IQC that underlies the nonlinear over-approximation. For PWL systems, Carathéodory solutions are absolutely continuous, so Theorem 16 is not affected; the concern is specific to reset systems. The paper explicitly assumes u ∈ U implies x, y ∈ L2 and states that ẋ ∈ L2, but does not state any jump-aware condition that would restore lim_{t→∞} x(t)=0. The pulse-train example shows the cited theorem's hypotheses are not satisfied by general right-continuous piecewise-AC signals, so the proof as written has a gap. Since the reader's conditional verdict already reflects this concern and the paper otherwise contains independent support (exact LTI results, worked examples, and a clear LMI framework), I recommend no change to the verdict.","tokens_in":20908,"tokens_out":38862,"duration_ms":548236,"concrete_test":"Verify the limit step in Theorem 15 by checking whether the cited Desoer–Vidyasagar theorem applies to right-continuous piecewise-AC signals: take f(t)=1 on [n, n+n⁻²] and 0 otherwise; f ∈ L2, f′ = 0 a.e., yet f(t) does not converge to 0, so the cited theorem cannot be invoked merely from x, ẋ ∈ L2 for reset trajectories. Then test whether a reset system of the form (42) can realize a trajectory with x ∈ L2, ẋ ∈ L2 a.e., and no limit as t→∞; if such a trajectory is admissible, Theorem 15 needs an extra hypothesis or a jump-aware proof. If no such reset trajectory exists, the gap is only in the citation and can be closed by a specialized argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 15, inequality (48) gives W(x(T)) ≤ ∫₀ᵀ ξᵀΘξ dt for every T. To pass to the soft IQC (49), the proof needs lim_{T→∞} W(x(T)) = 0, which it obtains from 'u ∈ U ⊂ L2 implies x ∈ L2, hence ẋ ∈ L2' and the citation [14, p.237]. The cited result is stated for absolutely continuous functions. Reset-system trajectories in (42) are only right-continuous and piecewise absolutely continuous, with jumps at reset instants. For such signals, membership in L2 together with a.e. L2 derivative does not force a limit at infinity: the function f(t)=1 on intervals [n, n+n⁻²] and 0 elsewhere is in L2, has f′ = 0 a.e., but has no limit as t→∞. Thus the proof has a genuine gap unless an additional jump-aware argument is supplied, e.g., summability of reset increments or a direct dissipation-based limit argument. Without this, the conclusion SG_U(HR) ⊆ S(Π) is not established for reset trajectories satisfying only the stated L2 conditions. This is load-bearing because it is the step that converts a finite-horizon dissipation inequality into the global IQC defining the region S(Π).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes an LMI-based dissipativity framework for computing scaled graphs of dynamical systems. It first establishes a link between quadratic integral quadratic constraints of the form (13) and disk/half-plane regions in the complex plane (Lemma 4), then uses this to construct over-approximations of scaled graphs by intersecting the regions generated by feasible matrices. For normal LTI systems, Theorem 12 claims exactness of the intersection when all disk centers on the real line are used. The framework is extended to reset systems (Theorem 15) and piecewise linear systems (Theorem 16) by combining the LMI conditions with S-procedure arguments, and to hard scaled graphs by adding a positive-semidefinite storage constraint. The paper closes with numerical examples, including feedback interconnection tests based on a scaled-graph separation theorem.","tokens_in":21243,"tokens_out":12310,"duration_ms":148557,"significance":"If the results hold, the paper gives a practical convex route to scaled-graph construction for several system classes, and the exactness result for normal LTI systems is a genuinely novel contribution that goes beyond transfer-function-based constructions. The paper is self-contained, uses standard tools (KYP lemma, S-procedure, supporting hyperplane theorem), and does not fit free parameters to benchmark outputs. The comparison with the reset-system method of [35] suggests a substantial reduction of conservatism. However, the proof of the reset-system theorem contains a genuine gap in the passage from a finite-horizon dissipation inequality to the global IQC, so the nonlinear extension is not yet established as stated.","major_comments":[{"comment":"The proof invokes [14, p. 237] to conclude lim_{t→∞} x(t) = 0 from x, ẋ ∈ L2, and then takes T → ∞ in (48) to obtain the soft IQC (49). The cited result is for absolutely continuous functions, but reset trajectories in (42) are only right-continuous and piecewise absolutely continuous, with jumps at reset instants. For such signals, membership in L2 together with an L2 a.e. derivative does not force a limit at infinity: for example, f(t)=1 on intervals [n, n+n^{-2}] and f(t)=0 elsewhere satisfies f, f′ ∈ L2 with f′ = 0 a.e. but has no limit as t→∞. Therefore W(x(T)) in (48) need not vanish along reset trajectories, and the step from (48) to (49) is not justified. This is load-bearing because (49) is exactly the IQC used to apply Lemma 4 and conclude SG_U(H_R) ⊆ S(Π). The authors should either add a reset-aware convergence argument (e.g., summability of reset increments or a direct dissipation-based limit) or impose an additional assumption that guarantees lim_{t→∞} x(t) = 0 along all admissible reset trajectories.","section":"Section 5.1, Theorem 15, Eqs. (48)-(49)"},{"comment":"The proof of exactness applies Lemma 11, which assumes that the closed convex set has nonempty interior. However, f_BK(SG(H_L)_+) can be a curve segment with empty interior: for the first-order system in Remark 14 the scaled graph is a circle, whose image under the Beltrami-Klein map is a line segment. Thus the hypothesis of Lemma 11 is not verified for cases that the theorem is intended to cover. The conclusion of (36) is still true by the standard supporting-hyperplane theorem for general closed convex sets, so the argument is repairable, but as written the cited lemma does not support the claim.","section":"Theorem 12, Section 4.3, proof around Eq. (36)"}],"minor_comments":[{"comment":"In the text introducing the Beltrami-Klein mapping, “maping” should be “mapping”.","section":"Section 2.3"},{"comment":"In the sentence “not limited to te systems considered,” “te” should be “the”.","section":"Section 5.2"},{"comment":"The abstract states that the approach “is shown to be exact for specific linear time-invariant systems,” but Theorem 12 establishes equality with the closure of the scaled graph and requires solving infinitely many LMIs (Λ_i = R and Λ_e = R). Remark 13 acknowledges this, but the abstract would benefit from the same qualification to avoid overstating the algorithmic exactness.","section":"Abstract and Remark 13"},{"comment":"The comparison with the method of [35] would be more reproducible if the authors specified whether the same grid sets Λ_i and Λ_e and the same computational effort were used for both methods.","section":"Example 2 and Figure 7"}],"recommendation":"major_revision","confidential_remarks":"The reset-system proof gap in Theorem 15 is the main obstacle to accepting the nonlinear extension. The LTI exactness result is strong and the reset gap appears repairable by adding a reset-aware convergence assumption or a more careful argument, so I would encourage a revision rather than rejection. The paper's comparison with [35] appears fair and is not a concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper is worth your time. The main new idea is to compute over-approximations of scaled graphs by intersecting regions generated from IQCs, rather than taking unions as in [35]. The exterior-disk parameterization Π(+1, λc, r) is a simple but effective trick: it produces non-convex over-approximations and gives noticeably tighter results in the reset-system example. The LMI problems (L1, L2 and the nonlinear variants) are clean and should be easy for others to implement. The strongest result is Theorem 12, showing exactness for normal LTI systems when the grid covers the real line. That is a real contribution.\n\nThe soft spots are two proof gaps, one minor and one load-bearing.\n\nFirst, Theorem 12 invokes Lemma 11, which requires the convex set to have nonempty interior. For a first-order system, SG is a circle, a curve with empty interior, so the lemma does not apply as stated. This is easily fixed: the intersection of all supporting half-planes of a closed convex set is the set itself even when the interior is empty. The result is very likely correct, but the proof as written needs that patch.\n\nSecond, and more seriously, the proof of Theorem 15 for reset systems uses Desoer–Vidyasagar [14, p.237] to conclude lim_{t→∞} x(t)=0 from x, ẋ ∈ L2. That theorem holds for absolutely continuous functions. Reset trajectories are only right-continuous and piecewise absolutely continuous. The difference matters: a function that is 1 on intervals [n, n+n^{-2}] and 0 elsewhere lies in L2 and has derivative 0 a.e., but has no limit at infinity. So the step from the finite-horizon inequality (48) to the soft IQC (49) is not justified. This is load-bearing because that IQC defines the region S(Π). The theorem may still be true under extra assumptions, such as convergence of all admissible trajectories, but those are not stated. The PWL theorem (Theorem 16) does not have this problem, since Carathéodory solutions are absolutely continuous.\n\nThe paper also ships no code or solver configuration, so exact reproduction takes some work, but the examples are detailed enough to be creditable.\n\nOverall: this is a serious contribution that deserves peer review. I would ask the authors to close the reset-system gap, either by proving convergence from the stated assumptions or by adding an explicit convergence condition, and to fix the supporting-half-plane argument. With those changes, the paper should be publishable.\n\nBest,","headline":"Solid LMI framework for scaled graphs with a clean LTI exactness result; the reset-system proof has a real gap in the limit argument.","tokens_in":21754,"tokens_out":7384,"would_cite":true,"duration_ms":80745,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C10","93C30","93D05","90C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"For normal LTI systems, the scaled graph—a nonlinear generalization of the Nyquist plot—can be computed exactly by intersecting LMI-certified disk and half-plane regions, and the same LMI framework gives tight over-approximations for…","keywords":["scaled relative graphs","scaled graphs","linear matrix inequalities","integral quadratic constraints","dissipativity","reset systems","piecewise linear systems","graphical stability analysis"],"falsifier":"For a normal stable controllable LTI system, compute the true scaled graph from the transfer-function spectrum by Theorem 8 and compare it with the L1/L2 intersection on a dense grid of real centers; any residual area outside the true graph refutes exactness. For a reset system, find a single input $u\\in L_2$ whose trajectory does not converge to zero (for example, a well-posed non-Zeno trajectory with persistent resets); if such an input exists, the limit step leading to the soft IQC (49) fails and the over-approximation argument collapses.","tokens_in":20740,"feed_emoji":"📈","tokens_out":11046,"duration_ms":115707,"temperature":0.7,"pith_summary":"The scaled graph of a system records, as points in the complex plane, the gain and phase of every possible input–output pair; for nonlinear systems this is normally impossible to compute exactly because it requires probing uncountably many inputs. This paper shows that the scaled graph can be approximated—and for normal linear time-invariant systems reproduced exactly—by solving linear matrix inequalities (LMIs), one family of inequalities per candidate disk or half-plane in the complex plane. The bridge is dissipativity: each quadratic supply rate that the system satisfies corresponds to an integral quadratic constraint, and that constraint corresponds to a region in the complex plane containing the scaled graph. Intersecting all such regions tightens the over-approximation, and the authors prove that for normal, stable, controllable LTI systems with all real disk centers allowed, the intersection is exactly the closure of the scaled graph. The same construction, with additional S-procedure multipliers, yields much tighter over-approximations for reset and piecewise-linear systems, making Nyquist-style graphical feedback analysis practical for those classes.","feed_headline":"LMI regions reconstruct the scaled graph exactly","feed_subtitle":"The same dissipativity-based construction gives tight approximations for reset and piecewise-linear systems.","key_machinery":"The computed object is the scaled graph $SG_U(H)=\\{\\rho(u,y)e^{\\pm j\\theta(u,y)}: u\\in U\\setminus\\{0\\},\\, y\\in H(u)\\}$, where $\\rho=\\|y\\|/\\|u\\|$ and $\\theta=\\arccos(\\langle u,y\\rangle/(\\|u\\|\\|y\\|))$. The workhorse is the matrix family $\\Pi(\\sigma,\\lambda_c,r)=\\sigma\\begin{bmatrix}1 & -\\lambda_c\\\\ -\\lambda_c & \\lambda_c^2-r^2\\end{bmatrix}$ with $\\sigma\\in\\{\\pm1\\}$, which generates exactly the interior disks $(\\sigma=-1)$ and their exteriors $(\\sigma=+1)$ centered on the real axis, together with half-planes as limiting cases. Each $\\Pi$ defines a quadratic supply rate, and Lemma 4 says the system satisfies the corresponding IQC exactly when the scaled graph lies inside $S(\\Pi)$. Feasibility of the associated KYP-style LMI in the storage matrix $P$ then certifies the region, and intersecting all certified regions produces the over-approximation. The exactness proof is carried by the Beltrami-Klein map $f_{BK}(z)=\\frac{(\\bar z-j)(z-j)}{1+|z|^2}$, which sends hyperbolic geodesic arcs to Euclidean line segments, so that hyperbolic convexity becomes Euclidean convexity and supporting half-planes in the disk correspond precisely to disk and exterior regions in the complex plane.","core_discovery":"The paper's central claim is that scaled graphs do not have to be probed input-by-input: they can be built from the feasible set of a semidefinite program. Lemma 4 establishes that a stable system satisfies the IQC $\\int_0^\\infty \\begin{bmatrix} y(t)\\\\ u(t) \\end{bmatrix}^\\top (\\Pi\\otimes I_n) \\begin{bmatrix} y(t)\\\\ u(t) \\end{bmatrix}dt\\geq 0$ if and only if its scaled graph lies in the region $S(\\Pi)=\\{z\\in\\mathbb{C}: [z;\\,1]^*\\Pi[z;\\,1]\\geq 0\\}$, which for $\\det\\Pi<0$ is a disk, the exterior of a disk, or a half-plane. Theorem 9 turns this into an LMI certificate: for an LTI system with Hurwitz $A$, existence of $P$ satisfying the KYP-style LMI implies the IQC, and under controllability all three statements—LMI feasibility, IQC, and scaled graph containment—are equivalent. The exactness theorem for normal LTI systems then uses the Beltrami-Klein mapping to convert the hyperbolic convex hull of the transfer-function spectrum into a Euclidean convex set, whose supporting half-planes correspond exactly to disk and exterior regions feasible for the LMIs; with $\\Lambda_i=\\Lambda_e=\\mathbb{R}$ the intersection of all such regions is the closure of the scaled graph. For reset and piecewise-linear systems, Theorems 15 and 16 certify the same regions using jump and flow-set multipliers, giving over-approximations that the examples show are much tighter than the previous union-based approach.","pith_inferences":["A natural next test, not run in the paper, is whether the exactness theorem extends to non-normal LTI systems; comparing the L1/L2 intersection against the Beltrami-Klein hull of the full numerical range would show whether normality is essential.","The exact LMI-intersection representation suggests an adaptive algorithm: instead of dense grids of centers $\\lambda_c$, one can place new disks where the current over-approximation most exceeds a sampled under-approximation, achieving certified $\\varepsilon$-tightness with far fewer LMI solves.","The reset-system proof invokes a decay theorem for absolutely continuous functions on right-continuous trajectories with jumps; a dwell-time or jump-bound condition may be needed to make the soft scaled graph argument fully rigorous, and a counterexample input with persistent resets would settle it.","Because separation of scaled graphs is a sufficient stability condition, exact computation for LTI components opens a loop-shaping design route: choose controller parameters so that the controller's scaled graph avoids the inverse scaled graph of the plant, with the LMI intersection as a fast graphical check."],"forward_implications":["For normal, stable, controllable LTI systems, the scaled graph can be computed exactly by solving the L1 and L2 LMI problems over all real centers, eliminating conservatism in that class.","The same LMI machinery produces significantly tighter over-approximations for reset control systems than the previous union-based method, and extends to piecewise-linear systems.","Feedback stability and L2-gain bounds for mixed interconnections (LTI with resets, resets with piecewise-linear systems) can be verified graphically through scaled-graph separation, with robustness margins read directly from the plots.","Hard scaled graphs for unbounded systems such as integrators are obtained by the same procedure with the extra constraint $P\\succeq 0$, so the framework covers systems outside $L_2$.","The construction is not tied to continuous-time LTI structure and can be adapted to any system class with LTI-like state-space descriptions, including discrete-time, LPV, and Lur'e systems."],"supporting_citations":[{"why":"Supplies the scaled graph and scaled relative graph definitions, hyperbolic convexity, and the graphical analysis context that the paper extends.","marker":"[9]"},{"why":"Defines soft and hard scaled graphs and provides the separation theorem (Theorem 1) used to turn scaled graph approximations into feedback stability and L2-gain conclusions.","marker":"[12]"},{"why":"Provides the integral quadratic constraint framework whose regions S(Pi) are connected to scaled graphs in Lemma 4.","marker":"[24]"},{"why":"Gives the representation of the scaled graph of a normal LTI system as the Beltrami-Klein h-hull of its spectrum, which the exactness proof builds on.","marker":"[26]"},{"why":"Supplies the Kalman-Yakubovich-Popov lemma that makes LMI feasibility equivalent to the IQC, and hence to scaled graph containment, under controllability.","marker":"[28]"},{"why":"Earlier union-based over-approximation of scaled graphs for reset systems; the paper's intersection-based method is compared against it and reduces its conservatism.","marker":"[35]"},{"why":"Foundational reference for dissipativity with storage functions, the notion used to derive the IQC conditions in Theorems 15 and 16.","marker":"[38]"},{"why":"Provides the decay result (square-integrable state and derivative imply the state tends to zero) that converts the dissipation inequality into the soft IQC.","marker":"[14]"},{"why":"Supplies the supporting-half-plane intersection lemma used in the exactness proof to pass from Euclidean convex hulls to disk and exterior regions.","marker":"[5]"}],"fun_headline_variants":["Scaled graph from LMI: exact for LTI, tight for rest","Dissipativity constructs scaled graph without probing inputs","LMI feasibility yields scaled graph exactly for LTI systems","From IQC to scaled graph: a unified construction","LMI-based scaled graph covers LTI, impulsive, and PWL"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For all nonlinear, reset, and piecewise-linear results, the argument assumes that every admissible input produces square-integrable state and derivative trajectories, so the state decays to zero and the indefinite storage term can be discarded; for reset systems this decay step is applied to right-continuous trajectories with jumps, where the standard theorem does not directly apply.","fun_headline_variants_meta":{"raw":{"variants":["Scaled graph from LMI: exact for LTI, tight for rest","Dissipativity constructs scaled graph without probing inputs","LMI feasibility yields scaled graph exactly for LTI systems","From IQC to scaled graph: a unified construction","LMI-based scaled graph covers LTI, impulsive, and PWL"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000303,"raw_usage":{"total_tokens":1808,"prompt_tokens":1076,"completion_tokens":732,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":692,"completion_tokens_details":{"reasoning_tokens":647}},"tokens_in":692,"tokens_out":732,"duration_ms":8060,"temperature":1.0,"reasoning_tokens":647,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:20:17.819278+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a normal stable controllable LTI system, compute the true scaled graph from the transfer-function spectrum by Theorem 8 and compare it with the L1/L2 intersection on a dense grid of real centers; any residual area outside the true graph refutes exactness. For a reset system, find a single input $u\\in L_2$ whose trajectory does not converge to zero (for example, a well-posed non-Zeno trajectory with persistent resets); if such an input exists, the limit step leading to the soft IQC (49) fails and the over-approximation argument collapses.","supporting_citations":[{"cited_title":"Chaffey, F","cited_arxiv_id":null,"evidence_quote":"Supplies the scaled graph and scaled relative graph definitions, hyperbolic convexity, and the graphical analysis context that the paper extends."},{"cited_title":"Megretski and A","cited_arxiv_id":null,"evidence_quote":"Provides the integral quadratic constraint framework whose regions S(Pi) are connected to scaled graphs in Lemma 4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Kalman-Yakubovich-Popov lemma that makes LMI feasibility equivalent to the IQC, and hence to scaled graph containment, under controllability."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier union-based over-approximation of scaled graphs for reset systems; the paper's intersection-based method is compared against it and reduces its conservatism."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Foundational reference for dissipativity with storage functions, the notion used to derive the IQC conditions in Theorems 15 and 16."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the decay result (square-integrable state and derivative imply the state tends to zero) that converts the dissipation inequality into the soft IQC."}],"review_version":1}