{"id":"57f019e9-0d02-4f9d-a4a5-731a36d7bee9","arxiv_id":"2606.12955","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":3,"one_line_summary":"A data-driven method embeds a p-copy internal model in a dynamic controller and uses a noise-robust SDP to synthesize gains that make the closed-loop exponentially contractive, ensuring almost periodic steady-state error with zero Fourier-Bohr coefficients at target frequencies and bounded residual ","lead":"The paper develops a data-driven controller using an internal model and semidefinite programming to achieve frequency-selective output regulation for unknown nonlinear systems under almost periodic disturbances. A smart generalist might read it because it shows how to design regulators that cancel specific frequency errors without identifying the full plant model first.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Linear parameterization over prescribed dictionary underpins data-driven SDP and all subsequent contraction/Fourier-Bohr claims","rationale":"The reader's weakest_assumption is exactly the modeling choice that makes every later step (data-driven SDP, contraction certificate, Fourier-Bohr elimination of embedded frequencies, Parseval residual bound) possible. No other internal gap appears more central once this premise is granted.","tokens_in":1794,"tokens_out":307,"duration_ms":15411,"concrete_test":"Re-run the SDP and closed-loop simulation of §5 with a 5 % additive perturbation to one dictionary function (e.g., replace sin(x) by sin(x)+0.05x); if the certified contraction rate drops below the value used in the almost-periodicity proof or the frequency-selective error bound is violated, the assumption is load-bearing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The entire construction begins from the assumption that the unknown nonlinear plant, input, output, and exosignal maps are each linear in a user-chosen dictionary whose functional form is known exactly. This lets the authors replace the unknown coefficient matrices by data matrices inside a single SDP whose feasibility certifies contraction of the augmented closed-loop vector field. Without this structural assumption the SDP cannot be written, the internal-model embedding cannot be data-driven, and the contraction-to-almost-periodicity argument has no starting point. The paper states the assumption explicitly but provides no robustness margin or dictionary-misspecification analysis.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript develops a data-driven frequency-selective output regulation scheme for continuous-time nonlinear plants whose vector fields, input, output, and exosignal maps are each linear in a user-chosen nonlinear dictionary, with all coefficient matrices unknown. A p-copy internal model is embedded in a dynamic controller; measured input-state-output data are substituted directly into a noise-robust SDP whose feasibility certifies exponential contraction of the augmented closed-loop vector field on a prescribed operating set. Contraction theory together with Fourier-Bohr analysis is then invoked to establish existence of a unique bounded attracting trajectory that is almost periodic, whose regulation-error components at the prescribed exosystem frequencies vanish, and whose residual spectral energy satisfies an explicit Parseval-type time-averaged bound. The approach is illustrated on a quadrotor with cable-suspended payload.","tokens_in":1944,"tokens_out":629,"duration_ms":24775,"significance":"If the central claims hold, the work is significant because it supplies the first data-driven internal-model design that simultaneously guarantees contraction, almost-periodicity of the steady-state error, and selective elimination of known frequencies without requiring model identification or exosignal amplitude/phase measurements. The explicit combination of contraction theory with Fourier-Bohr analysis to obtain a quantitative residual-energy bound, together with the reproducible SDP formulation from data, constitutes a clear technical advance for nonlinear output regulation under almost-periodic disturbances.","major_comments":[{"comment":"Abstract and the modeling assumption stated therein: the entire data-driven SDP construction, the contraction certificate, and the subsequent Fourier-Bohr claims rest on the plant being exactly linearly parameterized over a user-prescribed dictionary whose functional form is known a priori; no robustness margin, dictionary-misspecification bound, or sensitivity result is supplied, rendering the feasibility of the SDP and the contraction property sensitive to this modeling choice.","section":"Abstract / modeling assumption"},{"comment":"The derivation of the Parseval-type time-averaged energy bound on unmodeled spectral components (the step that converts contraction into the frequency-selective guarantee): the bound is asserted to follow from the contraction rate and the diameter of the operating set, yet the manuscript does not display the explicit dependence of the constant on these quantities or verify that the bound remains useful when the operating set is enlarged to accommodate realistic initial conditions.","section":"Proof combining contraction theory with Fourier-Bohr analysis"}],"minor_comments":[{"comment":"The abstract refers to a 'noise-robust semidefinite program' but the precise noise model (additive, multiplicative, bounded-energy, etc.) and the corresponding robustness certificate are not restated in the main text; a short paragraph clarifying the noise assumption would improve readability.","section":null},{"comment":"Notation for the data matrices that replace the unknown coefficient matrices inside the SDP should be introduced once, with a clear mapping to the original plant matrices, to avoid repeated re-definition across sections.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the constructive and detailed review. We address each major comment below.","responses":[{"response":"The manuscript explicitly restricts attention to the class of systems admitting an exact linear parameterization over a user-prescribed dictionary; this modeling choice is stated in the abstract and formalized in Section II. The data-driven SDP and contraction certificate are derived under this exact assumption, which permits direct use of measured data without identification. While a general robustness or sensitivity analysis with respect to dictionary misspecification would be a valuable extension, it lies outside the scope of the present work. In the revised manuscript we will insert a clarifying paragraph in the introduction and conclusions that reiterates the modeling assumption and discusses practical dictionary selection.","revision_made":"partial","referee_comment":"[Abstract / modeling assumption] Abstract and the modeling assumption stated therein: the entire data-driven SDP construction, the contraction certificate, and the subsequent Fourier-Bohr claims rest on the plant being exactly linearly parameterized over a user-prescribed dictionary whose functional form is known a priori; no robustness margin, dictionary-misspecification bound, or sensitivity result is supplied, rendering the feasibility of the SDP and the contraction property sensitive to this modeling choice."},{"response":"We agree that the explicit functional dependence should be displayed. The time-averaged residual-energy bound is obtained by combining the exponential contraction rate λ of the closed-loop vector field with the diameter D of the prescribed operating set, yielding an explicit constant of the form C(λ,D) that appears in the Parseval-type estimate. In the revised version we will state this dependence directly in the statement of the relevant theorem and add a short remark illustrating the scaling of C with respect to larger D, thereby confirming that the bound remains informative for operating sets sized to accommodate realistic initial conditions.","revision_made":"yes","referee_comment":"[Proof combining contraction theory with Fourier-Bohr analysis] The derivation of the Parseval-type time-averaged energy bound on unmodeled spectral components (the step that converts contraction into the frequency-selective guarantee): the bound is asserted to follow from the contraction rate and the diameter of the operating set, yet the manuscript does not display the explicit dependence of the constant on these quantities or verify that the bound remains useful when the operating set is enlarged to accommodate realistic initial conditions."}],"tokens_in":1581,"tokens_out":498,"duration_ms":17890,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper shows how to embed a p-copy internal model in a controller for unknown nonlinear plants, then use measured data in an SDP to make the closed loop contractive on a chosen set. From there contraction plus Fourier-Bohr analysis gives an almost-periodic steady-state trajectory whose error has zero Fourier coefficients at the prescribed exosystem frequencies and a Parseval-type bound on the rest. The quadrotor-with-payload example illustrates it on a physical system.\n\nThe combination of data-driven synthesis, frequency-selective objective, and contraction proof for almost-periodic exosignals looks new relative to the internal-model and data-driven regulation papers cited. It cleanly sidesteps full model identification and explicit exosignal amplitude measurement, which is useful when those are hard to get.\n\nThe load-bearing piece is the assumption that every map (plant, input, output, exosignal channel) is linear in a user-chosen nonlinear dictionary whose form is known exactly. The SDP is built directly from that structure; without it the data matrices and the contraction certificate do not exist. The paper states the assumption but supplies no misspecification analysis or robustness margin, so the result is conditional on the dictionary being adequate. The operating set is also prescribed in advance, which shifts some design burden to the user.\n\nThe work is aimed at people already working on data-driven nonlinear regulation or contraction-based control. It is coherent on its own terms and the cited tools (contraction theory, Fourier-Bohr) are applied in a standard way, so it deserves a serious referee even though the dictionary assumption will need scrutiny in review.","headline":"Data-driven frequency-selective regulation via contraction and internal model, but the linear dictionary assumption carries the whole thing.","tokens_in":2419,"tokens_out":391,"would_cite":false,"duration_ms":14426,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A data-driven controller with embedded internal model eliminates prescribed frequency components from the steady-state regulation error of unknown nonlinear systems under almost periodic exosignals.","keywords":["output regulation","data-driven control","almost periodic signals","internal model","contraction theory","nonlinear systems","frequency-selective regulation","semidefinite programming"],"falsifier":"A simulation or experiment in which the closed-loop regulation error exhibits nonzero Fourier-Bohr coefficients at any embedded exosystem frequency, or in which the time-averaged energy of the residual spectrum exceeds the derived bound, would falsify the central claim.","tokens_in":2704,"feed_emoji":"","tokens_out":742,"duration_ms":17290,"temperature":0.7,"pith_summary":"The paper shows how to achieve frequency-selective output regulation for nonlinear plants whose dynamics are linearly parameterized over a known dictionary but whose coefficient matrices remain unknown. It embeds a p-copy internal model into a dynamic controller, represents the unknown matrices directly from measured data, and solves a noise-robust semidefinite program to make the augmented closed-loop vector field exponentially contractive on a chosen operating set. Contraction guarantees a unique bounded attracting trajectory whose regulation error is almost periodic; the Fourier-Bohr coefficients at the embedded exosystem frequencies vanish while the time-averaged energy of all other spectral components satisfies an explicit Parseval-type bound. This matters for applications such as aerial vehicles because it removes the need for plant identification or direct measurement of exosignal amplitudes and phases.","feed_headline":"Data-driven controller zeros selected frequencies in regulation error","feed_subtitle":"Without plant identification, measured data and a semidefinite program enforce almost periodic error with eliminated exosystem components an","key_machinery":"The noise-robust semidefinite program that computes the controller gain from input-output data to enforce exponential contractivity of the augmented system containing the p-copy internal model.","core_discovery":"Embedding a p-copy internal model yields an augmented nonlinear system whose unknown constant matrices are expressed directly by measured data; a semidefinite program then synthesizes the controller gain so that the closed-loop vector field is exponentially contractive, which implies existence and uniqueness of a bounded attracting trajectory that is almost periodic, whose regulation error has zero Fourier-Bohr coefficients at the embedded frequencies, and whose residual spectral components obey a Parseval-type time-averaged energy bound.","pith_inferences":["The same contraction-plus-Fourier-Bohr argument might be reused to bound regulation error under exosignals whose frequencies drift slowly within a known band.","Replacing the fixed nonlinear dictionary with an online-updated one would require only that the semidefinite program be solved periodically while preserving the contraction margin.","The Parseval-type bound supplies a concrete performance certificate that could be compared against classical internal-model designs once the plant matrices are identified."],"forward_implications":["The closed-loop system possesses a unique bounded and globally attracting steady-state trajectory on the prescribed operating set.","The regulation error trajectory is almost periodic and contains none of the embedded exosystem frequencies.","The time-averaged energy of all unmodeled spectral components of the error is bounded by a quantity derived from the contraction rate and the internal-model dimension.","The design applies directly to plants whose input, output, and exosignal channels also contain unknown constant matrices."],"fun_headline_variants":["Data-driven internal model cancels select frequencies in error","Controller from data makes regulation error zero at exosystem freqs","Frequency selective output regulation for nonlinear systems via data","Contractive closed loop eliminates embedded frequencies from error"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The plant dynamics admit a linear parameterization over a prescribed nonlinear dictionary while all coefficient matrices remain unknown constants.","fun_headline_variants_meta":{"raw":{"variants":["Data-driven internal model cancels select frequencies in error","Controller from data makes regulation error zero at exosystem freqs","Frequency selective output regulation for nonlinear systems via data","Contractive closed loop eliminates embedded frequencies from error"]},"model":"grok-4.3","cost_usd":0.006357,"raw_usage":{"total_tokens":3023,"prompt_tokens":745,"num_sources_used":0,"completion_tokens":60,"cost_in_usd_ticks":63574500,"prompt_tokens_details":{"text_tokens":745,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2218,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":745,"tokens_out":60,"duration_ms":17027,"temperature":1.0,"reasoning_tokens":2218,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T05:59:51.288040+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A simulation or experiment in which the closed-loop regulation error exhibits nonzero Fourier-Bohr coefficients at any embedded exosystem frequency, or in which the time-averaged energy of the residual spectrum exceeds the derived bound, would falsify the central claim.","supporting_citations":[],"review_version":1}