{"id":"15dfbc01-3193-47f4-9f56-b9ffc6cea97d","arxiv_id":"2505.09255","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A data-driven internal model controller achieves zero or kth-order asymptotic output regulation for unknown linear, nonlinear, and multi-agent systems without solving regulation equations.","lead":"This paper designs controllers for unknown systems that must track a reference or reject a disturbance, using only a batch of measured data. It achieves zero tracking error for linear systems and approximate tracking for nonlinear systems, by turning the task into a stabilization problem that a standard optimization routine can solve.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption 5 requires a known bound on EξV+D, but Eξ contains the unknown plant matrices E,F; the paper gives no data-only procedure to select Δ, so the zero-error claim rests on an unverifiable prior.","rationale":"The reader identified Assumption 5 as the weakest point, and this pass agrees. I checked whether the LMI argument can be repaired from within the paper. The proof of Theorem 2 is a direct application of Theorem 1 and is internally sound once a valid Δ is supplied, but no later section or remark supplies a construction of Δ from data; Remark 2 only mentions alternative noise models, not how to handle the EξV term. The numerical examples use known plant matrices, so they do not demonstrate a model-free procedure. A secondary issue is that Theorem 4 uses a unitary diagonalization of H that need not exist for non-normal Laplacians with complex eigenvalues, which would affect the MAS extension; I did not make that the headline because the nonconstructive Δ already weakens the single-agent result on which the main claim rests. The appropriate remedy is to state explicitly a known prior bound on Eξ (or on E,F), or to provide a data-based estimator for Eξ, and to note that zero-error tracking is conditional on such a bound. Since the mathematical implication 'valid Δ plus feasible LMI implies zero tracking' is not itself in question, the reader's conditional verdict remains appropriate.","tokens_in":20469,"tokens_out":11196,"duration_ms":128107,"concrete_test":"Re-run Example 1 in a controlled way: provide the designer only U-, Ξ-, Ξ+, the per-sample bounds ||d(τ)||∞ ≤ 0.01 and ||v(τ)||∞ ≤ 0.0025, G1, G2, and the exact declaration that no model matrices are known. Ask the designer to certify a Δ satisfying (20) and then solve LMI (23). If the only certified Δ is computed using E,F, or if no certifiable Δ yields a feasible LMI, then Assumption 5 is nonconstructive and the central model-free claim is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2's conclusion that controller (15) solves Problem 1 depends on showing Aξ+BξKξ is Hurwitz for the actual system. The proof places [Aξ,Bξ] in the data-consistent set Cξ in (21)-(22), but this inclusion is valid only if a matrix Δ satisfying (20) is available. The left-hand side (EξV+D)(EξV+D)^T involves Eξ = col(E, G2F), i.e., the unknown plant matrices E and F combined with the measured exosignal trajectory V. Thus Assumption 5 is not a data-only noise bound; it is a bound on a quantity depending on the unknown model. For finite data a Δ always exists (a large multiple of the identity), so the real content is that Δ must be known and certified. The paper gives no method to compute or verify such a Δ from U-, Ξ-, Ξ+ and declared noise/v bounds. If Δ is too small, the true pair lies outside Cξ and LMI (23) gives no guarantee for the actual closed loop; if Δ is too large, (23) may be infeasible. The examples do not settle this, because the authors know E and F when constructing the experiment. The same gap propagates through Theorems 3-5, so the claimed exact output regulation from noisy data is only as strong as an unstated prior bound on Eξ.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper addresses the output regulation problem for unknown linear time-invariant systems, kth-order nonlinear systems, and multi-agent systems using noisy input-state data. The main idea is to augment the plant with an internal model of the exosystem, transforming output regulation into a stabilization problem, and then to design the stabilizing gain via the robust data-driven LMI based on Petersen's lemma. The paper claims exact zero tracking error without solving output regulation equations. The claims are supported by four numerical examples and comparisons with a polytopic data-driven synchronization method.","tokens_in":20793,"tokens_out":12089,"duration_ms":121741,"significance":"The internal-model reformulation is an interesting and potentially impactful way to avoid the generally infeasible data-based output regulation equations, and the reduction to a convex LMI is elegant. If the core guarantees were fully data-driven, the paper would be a significant contribution to direct data-driven control. However, as discussed in the major comments, the main theorem is conditional on an unverifiable matrix bound that involves the unknown plant matrices, and a proof step in the multi-agent extension assumes diagonalizability of the graph matrix H. With these gaps repaired or the claims appropriately qualified, the paper would be a useful contribution for the data-driven control community.","major_comments":[{"comment":"Assumption 5 requires a known matrix Δ such that (EξV+D)(EξV+D)^T ⪯ ΔΔ^T, but Eξ = col(E, G2F) is defined in terms of the unknown plant matrices E and F. The paper does not provide a data-only procedure to construct or certify Δ; it only states existence. For finite data such a Δ always exists, so the substantive content is that a certified bound is available, and the examples do not establish this because E and F are known to the authors when the data are generated. If Δ is chosen too small, the true pair [Aξ,Bξ] may lie outside Cξ in (21), so LMI (23) gives no guarantee for the actual closed loop; if Δ is chosen too large, the LMI may be infeasible. The same issue propagates through Assumption 9 and Theorems 3, 4, and 5. The central claim of exact model-free output regulation is therefore not established as stated.","section":"III-A3, Eq. (20); Theorems 2–5"},{"comment":"The proof assumes a unitary T1 with T1 H T1^{-1} = diag{λ1,...,λN}. This is not guaranteed for the matrix H = L+Λ of a directed graph satisfying Assumption 7; H can be non-diagonalizable (for example, a directed path with the leader as root gives a Jordan block). The stability conclusion can still be obtained by using a Schur triangular form of H and the fact that a block-triangular matrix is Hurwitz if all diagonal blocks are Hurwitz, but the proof as written is not valid for general directed graphs. This also affects Theorem 5, which relies on Theorem 4.","section":"IV-A, proof of Theorem 4"}],"minor_comments":[{"comment":"Please clarify how the augmented state ξ = col(x,z) data are collected; in particular, specify that v is measured during the offline experiment and that z is obtained by simulating the internal model (15b) using the measured error e, since z is not a physical state.","section":"III-A3, around Eq. (19)"},{"comment":"The term \"-P\" on the left-hand side is not a consequence of Theorem 1's LMI (7b)/(23); either remove it or provide a separate argument for the stronger inequality.","section":"Theorem 2 proof, Eq. (24)"},{"comment":"The stabilizing gain obtained from LMI (55) is Y_i P_i^{-1} = λ_i Kξ_i, so the displayed inequality should involve λ_i B̄ξi Kξ_i (or a tilde gain) to match the definition Kξ_i := Y_i P_i^{-1}/λ_i.","section":"Theorem 4 proof, Eq. (56)"},{"comment":"The wording \"there exists some matrix Δ\" should be changed to \"a known matrix Δ is available such that ...\" to avoid a vacuous-existence interpretation that does not yield a constructor.","section":"Assumption 5"},{"comment":"There are several minor typos: \"diagraph\" in Problems 3 and 4 should be \"digraph\"; in Eq. (42b), the term \"h0x1x2/4\" appears to have missing or ambiguous parentheses; and in Property 2 of Problem 2, the expression `(e(t)− O(v(k+1)(t))` has an unbalanced parenthesis.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The main technical gap (Assumption 5) is within the scope of a revision if the authors can either construct Δ from data with an explicit prior or reframe the contribution as conditional on a certified bound. I would be willing to review a revised version. The H-diagonalization issue in Theorem 4 is a proof error that is easily fixed by triangularization; it does not by itself change the recommendation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The internal-model reformulation is a good trick: it converts the data-driven output regulation problem into a robust stabilization problem, sidestepping OREs entirely. And the paper mostly works, but the main theorem rests on an assumption that is not verifiable from data, and the MAS extension has a complex-eigenvalue bug.\n\nWhat is actually new: the paper shows that if you can bound the combined term EξV+D in the augmented state equation, then the standard Petersen-lemma LMI stabilizes the augmented system, and the internal model principle gives zero steady-state tracking. This genuinely goes beyond prior data-driven synchronization results that either required exact noise or accepted bounded tracking. The extensions to kth-order nonlinear regulation and to multi-agent systems are natural and plausible, and the examples are clean, including the comparison with the polytopic method in [36].\n\nThe soft spot is Assumption 5. The matrix Eξ contains the unknown plant matrices E and F, so (EξV+D)(EξV+D)^T cannot be bounded from data alone. The user must produce Δ by hand; too small and the true pair escapes the data-consistent set, too large and the LMI may be infeasible. The examples pick Δ with full knowledge of E and F. The claim of exact output regulation from noisy data is therefore only as strong as an unstated prior bound on the unknown system. The nonlinear version inherits this and makes it worse, since the higher-order remainder is also unknown. Theorem 4 divides the gain by the eigenvalues λi of H; when these are complex, Kξi becomes complex and the real controller is not implementable. That is a real bug, though probably fixable with a real Jordan form argument. The proofs of the nonlinear theorems are omitted, which is acceptable if the reduction is standard, but combined with Assumption 5 it makes the nonlinear claim feel unfinished. No code or data is shipped, which is a minor issue for a theory paper.\n\nWho this is for: data-driven control researchers, especially those working on output regulation and synchronization. The core idea is worth publishing once Assumption 5 is made constructible (e.g., by assuming a bound on the extended system matrices or on the exosignal contribution) and the complex eigenvalue issue is fixed. It deserves a serious referee.","headline":"A nice internal-model trick for data-driven output regulation, but the main guarantee rests on a noise bound that cannot be verified from data and the MAS section has a complex-eigenvalue bug.","tokens_in":21279,"tokens_out":3536,"would_cite":true,"duration_ms":36316,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Output regulation of unknown systems becomes stabilization of an augmented plant: one data-based LMI yields zero tracking error from noisy data, plus kth-order nonlinear and multi-agent extensions.","keywords":["data-driven control","output regulation","internal model principle","linear matrix inequality","noisy data","multi-agent systems","nonlinear output regulation","kth-order internal model"],"falsifier":"Simulate the robot example with known matrices, record the true combined disturbance-exosignal matrix $E_\\xi V+D$, and take $\\Delta$ from its largest singular value; then rerun the LMI with a deliberately smaller $\\Delta$ while holding the data fixed. If the LMI remains feasible and the closed-loop tracking error fails to converge to zero, Theorem 2's claim is contradicted, because the true augmented system no longer belongs to the data-consistent set on which the proof relies.","tokens_in":20321,"feed_emoji":"🎯","tokens_out":12096,"duration_ms":109779,"temperature":0.7,"pith_summary":"This paper claims that an unknown linear system can be made to track a reference and reject a disturbance with zero steady-state error using only noisy input-state data and a known description of the reference/disturbance generator. The proposed route is to embed an internal model of that generator into the controller, which turns output regulation into a stabilization problem, and then to compute the stabilizing gain from a single data-based linear matrix inequality (LMI). No output regulation equations are solved and no plant model is identified. The same idea is extended to give kth-order approximate tracking for a class of nonlinear systems and to solve linear and nonlinear cooperative output regulation for multi-agent systems.","feed_headline":"No plant model, zero tracking error: one LMI does output regulation","feed_subtitle":"Reformulating output regulation as stabilization lets noisy data alone drive exact tracking to zero.","key_machinery":"The load-bearing object is the augmented system $\\dot\\xi=A_\\xi\\xi+B_\\xi u+E_\\xi v$, $e=C_\\xi\\xi+Fv$, formed by appending the internal model $\\dot z=G_1 z+G_2 e$ to the unknown plant, with $\\xi=\\mathrm{col}(x,z)$. The paper's key move is the internal-model-principle reduction: finding a controller that solves output regulation is replaced by finding $K_\\xi$ that makes $A_\\xi+B_\\xi K_\\xi$ Hurwitz, so no output regulation equations are needed. On the data side, the workhorse is a robust data-driven stabilization result: from noisy trajectories of $\\xi$ one builds matrices $\\Psi$, $\\Upsilon$, $\\Sigma$ and a quadratic inequality describing the set of all augmented systems consistent with the data; feasibility of LMI (23) then yields a gain that stabilizes every system in that set, including the true plant.","core_discovery":"At the core is a reduction: for an unknown LTI plant whose reference and disturbance signals are generated by a known matrix $S$, the paper designs a controller $u=K_x x+K_z z$ with $\\dot z=G_1 z+G_2 e$, where $(G_1,G_2)$ is an $n_y$-copy internal model of $S$. Embedding the generator dynamics into the controller is the key step: by the internal model principle, whenever $A_\\xi+B_\\xi K_\\xi$ is Hurwitz, the output regulation equations are automatically solvable and $e(t)\\to 0$. The main theorem states that, under a persistency-of-excitation condition and a known energy bound on the combined noise and exosignal contribution, feasibility of the data-based LMI (23) produces $K_\\xi=YP^{-1}$ that stabilizes the true augmented system and solves the regulation problem. The same construction, using a $k$-fold internal model of the exosystem, solves the $k$th-order nonlinear output regulation problem, and distributed versions solve linear and nonlinear cooperative output regulation for heterogeneous multi-agent systems.","pith_inferences":["The practical bottleneck implied by this approach is the choice of the energy bound in Assumption 5; a data-only procedure that validates or adaptively tightens $\\Delta$ would make the zero-error guarantee usable in applications, but no such procedure is given.","Because the reduction is modular, other data-driven stabilization designs with different noise models or Lyapunov functions could be substituted for the robust stabilization step used here, potentially reducing conservativeness; the paper points toward this in Remark 6 but does not develop it.","A natural stress test not reported in the paper is to fix a known plant, collect data with a large exosignal, choose $\\Delta$ from a smaller exosignal, and check whether the tracking error still converges; the theory predicts it will not, because the true augmented system then falls outside the data-consistent set."],"forward_implications":["For unknown LTI systems, zero steady-state tracking error is achievable from noisy data by solving one convex LMI; the controller neither solves output regulation equations nor requires online measurements of the reference or disturbance signal.","For smooth nonlinear systems, a $k$-fold internal model makes the tracking error converge to a term of order $O(v^{k+1})$, so higher-order internal models give higher-order asymptotic tracking without identifying the nonlinear functions.","For heterogeneous linear multi-agent systems under a directed spanning-tree graph, the distributed protocol makes every agent's output track the exosystem output exactly; for nonlinear agents the same protocol gives $k$th-order cooperative output regulation.","Because the design reduces output regulation to robust stabilization, the data-based stabilization LMI is the only plant-dependent computation, and the resulting controller is obtained directly from offline noisy data rather than from a system identification step."],"supporting_citations":[{"why":"Supplies the robust data-driven stabilization theorem from noisy input-state data; the paper applies its LMI to the augmented system.","marker":"[16]"},{"why":"Provides the internal model principle and the lemma that stabilizing the augmented system implies solvability of the output regulation equations.","marker":"[1]"},{"why":"Supplies the kth-order internal model construction and the result that a stabilizing gain for the augmented system solves the kth-order nonlinear regulation problem.","marker":"[8]"},{"why":"Establishes persistency of excitation and the fundamental lemma, which justify the full-row-rank data assumption used to define the data-consistent system set.","marker":"[14]"},{"why":"The polytopic data-driven output synchronization method that only achieves bounded tracking; it serves as the baseline the paper compares against to show exact tracking.","marker":"[36]"},{"why":"An earlier data-driven output synchronization method that requires measurable and perfectly known process noise; the paper contrasts it with its bounded-noise assumption.","marker":"[35]"}],"fun_headline_variants":["Data-driven internal model: one LMI for zero tracking error","No model, no ORE: data-driven internal model regulates","One LMI solves output regulation without a plant model","Internal model reformulation: exact tracking from noisy data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result depends on the user first knowing a bound on the combined effect of the measurement noise and the reference/disturbance signal during data collection; because that effect involves the unknown plant, the bound cannot be verified from data alone, and if it is chosen too small the zero-error conclusion collapses.","fun_headline_variants_meta":{"raw":{"variants":["Data-driven internal model: one LMI for zero tracking error","No model, no ORE: data-driven internal model regulates","One LMI solves output regulation without a plant model","Internal model reformulation: exact tracking from noisy data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000533,"raw_usage":{"total_tokens":2585,"prompt_tokens":987,"completion_tokens":1598,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":603,"completion_tokens_details":{"reasoning_tokens":1531}},"tokens_in":603,"tokens_out":1598,"duration_ms":12595,"temperature":1.0,"reasoning_tokens":1531,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:36:27.124272+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the robot example with known matrices, record the true combined disturbance-exosignal matrix $E_\\xi V+D$, and take $\\Delta$ from its largest singular value; then rerun the LMI with a deliberately smaller $\\Delta$ while holding the data fixed. If the LMI remains feasible and the closed-loop tracking error fails to converge to zero, Theorem 2's claim is contradicted, because the true augmented system no longer belongs to the data-consistent set on which the proof relies.","supporting_citations":[{"cited_title":"Data-driven control via Petersen’s lemma,","cited_arxiv_id":null,"evidence_quote":"Supplies the robust data-driven stabilization theorem from noisy input-state data; the paper applies its LMI to the augmented system."},{"cited_title":"Huang, Nonlinear Output Regulation: Theory and Applications","cited_arxiv_id":null,"evidence_quote":"Provides the internal model principle and the lemma that stabilizing the augmented system implies solvability of the output regulation equations."},{"cited_title":"On a robust nonlinear servomechanism problem,","cited_arxiv_id":null,"evidence_quote":"Supplies the kth-order internal model construction and the result that a stabilizing gain for the augmented system solves the kth-order nonlinear regulation problem."},{"cited_title":"A note on persistency of excitation,","cited_arxiv_id":null,"evidence_quote":"Establishes persistency of excitation and the fundamental lemma, which justify the full-row-rank data assumption used to define the data-consistent system set."},{"cited_title":"Data-driven polytopic output synchronization from noisy data,","cited_arxiv_id":null,"evidence_quote":"The polytopic data-driven output synchronization method that only achieves bounded tracking; it serves as the baseline the paper compares against to show exact tracking."},{"cited_title":"Data-driven output synchronization of heterogeneous leader- follower multi-agent systems,","cited_arxiv_id":null,"evidence_quote":"An earlier data-driven output synchronization method that requires measurable and perfectly known process noise; the paper contrasts it with its bounded-noise assumption."}],"review_version":1}