{"id":"9bf7a931-0b67-4666-83f1-7ba33c39ff21","arxiv_id":"2506.06079","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A data-dependent LMI feasibility problem synthesizes a feedback regulator that, together with an exosystem-based internal model, achieves asymptotic reference tracking and disturbance rejection for nonlinear systems with unknown matrices.","lead":"A data-driven controller is designed that makes a nonlinear system track a reference and reject disturbances using only recorded data and known nonlinearity structure. The method converts the tracking problem into passivity conditions expressed as data-dependent matrix inequalities, guaranteeing asymptotic tracking whenever those inequalities are feasible.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's convergence proof assumes, without support, that the steady-state control residual is reproducible by the internal model; Example 1's sin(sin 2t) term shows this need not hold.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing gap: the existence of a bounded zero-error steady state of the augmented system, in particular the existence of ηss(t) with η̇=Sη and Ξ^Tηss(t)=u_ss(t)−KZ(x_ss(t)). My stress-test confirms this is the central unproven step in Theorem 2. The entire asymptotic tracking argument reduces to applying the incremental passivity inequality (8) with ζss(t) as the comparison trajectory; if ζss(t) does not exist, the storage function derivative bound and the subsequent Barbalat conclusion have no valid basis. The pendulum example (Example 1) is not a mere technicality: because Z(x) contains sin(x1) and x1,ss=sin(2t), the residual ρ(t) contains sin(sin 2t), which has spectral content beyond the internal model's S-generated subspace. No design freedom in Ξ is described that would make this residual reproducible; the theorem as stated claims success without any condition on Ξ. This is a genuine correctness risk, not merely an overstrong claim. I also considered whether the internal model (21) is incrementally passive for general S satisfying Assumption 1; the storage function V_IM requires S+S^T⪯0, which holds for the skew-symmetric S used in the examples but not for all S with eigenvalues on the imaginary axis (e.g., a nilpotent Jordan block). This is a second issue, but it can be repaired by choosing a skew-symmetric realization of the exosystem, whereas the ηss existence gap cannot be repaired without an additional assumption or a nonlinear internal model. Thus the dominant, load-bearing concern is the steady-state reproducibility of the control input residual. Because the reader's conditional verdict already accounts for this gap and asks for a design rule or additional assumption, my read does not change the verdict.","tokens_in":16938,"tokens_out":12590,"duration_ms":117693,"concrete_test":"For Example 1, compute the steady-state solution of the regulator equations (5) using the known model: x1,ss(t)=sin(2t), x2,ss(t)=2cos(2t)−cos(2t+π/3), and u_ss(t) from the pendulum dynamics. With the reported K=[−0.1290 −0.0132 1.0000] and Ξ=[1 0 1]^T, form ρ(t)=u_ss(t)−KZ(x_ss(t)). Check whether ρ(t) equals Ξ^Tη(t) for some η solving η̇=Sη with S=blockdiag([0,2;−2,0],0). Concretely, compute the least-squares fit of ρ(t) to the span of {sin(2t), cos(2t), 1} over a long horizon; the residual norm will not converge to zero because ρ contains sin(sin 2t) with harmonics at 4,6,..., proving the ζss used in Theorem 2's proof does not exist.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 2 requires a bounded zero-error steady-state trajectory of the augmented system (24), written as ζss(t)=(x(w), ηss(t)). For this to exist, ηss must solve η̇=Sη and satisfy Ξ^T ηss(t) = u(w)−KZ(x(w)), because e=0 and u=KZ(x)+Ξ^Tη−K̃e. Assumption 2 only guarantees a plant-level solution (x(w),u(w)) of the regulator equations (5); it does not guarantee that the residual ρ(t)=u(w)−KZ(x(w)) lies in the image of Ξ^T over the S-invariant subspace. The theorem states no condition on Ξ and gives no design rule. Example 1 makes the gap concrete: with r=sin(2t), x1,ss=sin(2t), so Z(xss) contains sin(sin 2t), whose Fourier expansion has frequencies 2,4,6,...; the internal model with S having eigenvalues ±2j and 0 can only generate frequencies 0 and 2. The required ηss therefore does not exist unless the coefficient of sin(sin 2t) in ρ is zero, which is not checked. Consequently, the storage function (25) is not evaluated along a valid comparison trajectory, and the Barbalat-based convergence argument in Theorem 2 is not supported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a data-driven output regulation scheme for a class of nonlinear systems with unknown matrices A, B, C, E, F, where the reference and disturbances are generated by a known linear exosystem. The controller consists of a static nonlinear feedback u = K Z(x) + Ξ^T η − K̂ e and a linear internal model η̇ = Sη − αΞ e. The gain K is obtained from offline data via LMIs that enforce incremental passivity of the closed-loop system with respect to the regulation error and a virtual input. The main theorem (Theorem 2) claims that if these data-dependent LMIs are feasible, then the regulator asymptotically drives the error to zero. A stabilization extension for non-zero equilibria with unknown equilibrium input is also given, along with three numerical examples.","tokens_in":17216,"tokens_out":18807,"duration_ms":170145,"significance":"If the main theorem were correct, the paper would offer a clean, decoupled, data-driven design for nonlinear output regulation with asymptotic guarantees, building on the model-based incremental passivity framework of Pavlov and Marconi and the data-based contraction conditions of Hu et al. The LMI conditions are derived from first principles, the data representation (17) is clearly explained, and the numerical examples are reproducible in structure. However, the central convergence proof contains an unproved existence assertion for a zero-error steady state of the augmented system, and the internal model's incremental passivity is only established for a restricted class of exosystem matrices. These gaps currently prevent the main theorem from being accepted as stated.","major_comments":[{"comment":"The proof of Theorem 2 asserts, without proof, that 'Under Assumption 2, system (24) admits a bounded solution ζss(t) = (x(w), ηss(t))'. Assumption 2 only guarantees a plant-level solution (x(w), u(w)) of the regulator equations (5). For the augmented system to have a zero-error bounded solution, there must exist ηss(t) such that η̇ss = S ηss and Ξ^T ηss(t) = u(w(t)) − K Z(x(w(t))). No condition on Ξ is stated in Theorem 2 and no design rule is given, so the comparison trajectory used in the incremental passivity inequality and in the Barbalat argument may not exist. This is load-bearing: the entire convergence proof relies on this trajectory. For polynomial x(w), u(w), the right-hand side generically contains harmonics that an internal model with only exosystem frequencies cannot produce. Example 1 does not rescue the theorem as stated, because although the specific K = [-0.1290 -0.0132 1.0000] cancels the sin(sin 2t) term in the steady-state residual, the theorem does not require or prove such cancellation.","section":"Section 4, Theorem 2"},{"comment":"The storage function V_IM = (1/(2α))||η1−η2||^2 yields a derivative with the term (1/α)(η1−η2)^T S(η1−η2), which must be nonpositive for the internal model to be incrementally passive. This holds only if S + S^T ⪯ 0, which for matrices with all eigenvalues on the imaginary axis is equivalent to S being skew-symmetric. Assumption 1 (all eigenvalues of S have zero real part) does not imply skew-symmetry; for example, S = [[0,2],[-0.5,0]] has eigenvalues ±i and S+S^T is indefinite. Thus the application of Lemma 1 to the internal model is unjustified for exosystems satisfying only Assumption 1. The authors should either assume S is skew-symmetric (or S+S^T ⪯ 0) or use a weighted storage function with a positive definite solution to the Lyapunov equation S^T P + P S = 0.","section":"Section 4, internal model (21), Lemma 1"},{"comment":"The representation W0 = Γ M0 with M(t) = [M1(t)^T ... 1_{q2}^T]^T is valid only for exosystem matrices that are block diagonal with 2x2 rotation blocks and a zero block, i.e., S is semi-simple with purely imaginary eigenvalues. Assumption 1 only states that all eigenvalues have zero real part, which permits nontrivial Jordan blocks (e.g., a double integrator S = [[0,1],[0,0]]). For such S, M0 does not capture polynomial-in-time signals, and the derivation X1 = A Z0 + B U0 + E W0 with W0 = Γ M0 in Lemma 5 fails. The theorem's assumptions therefore do not match the data representation used in the proofs of Theorems 1 and 2. The authors should either restrict Assumption 1 to semi-simple S or extend the data representation to handle Jordan blocks.","section":"Section 2, equations (4)-(6), Remark 1"},{"comment":"Assumption 2 requires x(w) and u(w) to be polynomials in w. For the pendulum example, the steady-state input obtained from the regulator equations is u(w) = 0.1(−3w1 + √3 w2) + sin(w1) + 0.1(1.5w2 + √3/2 w1) − 0.1, which contains sin(w1) and is not a polynomial in w. Thus Example 1 does not satisfy Assumption 2 and is not covered by Theorem 2. If the authors intend to allow non-polynomial steady-state solutions, Assumption 2 should be reformulated and the existence claim in Theorem 2 should be proved under the weaker assumption.","section":"Section 4, Example 1, Assumption 2"}],"minor_comments":[{"comment":"The interconnection should read '˜e = −e + v̂IM' rather than 'e = −e + v̂IM'.","section":"Section 4, equation (23)"},{"comment":"The statement 'Vaug_dot < 0 when ζ(t) ≠ ζss(t)' is not implied by the derived inequality Vaug_dot ≤ −e^T K̂ e ≤ 0, since e can vanish while ζ differs from ζss. This overstatement is not needed for the Barbalat argument.","section":"Section 4, proof of Theorem 2"},{"comment":"The matrix I is defined as [I_n 0_{n×(nZ−n)}] in Lemma 3, but later used in I^T X1 Y in Theorem 1 without a reminder of its dimensions; a brief restatement would improve readability.","section":"Section 3, Lemma 3 and Theorem 1"},{"comment":"The claim that Assumption 1 is 'slightly less restrictive than the one posed in [17]' is not substantiated; if the comparison concerns the eigenstructure used in the data representation, the statement should be qualified.","section":"Remark 1"},{"comment":"The figure captions do not list the controller parameters (α, K̂, Ξ) used in the simulations; including these values would aid reproducibility.","section":"Figures 2-4"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern is confirmed: the proof of Theorem 2 assumes, without support, the existence of a bounded zero-error steady state ζss(t) of the augmented system. This is a genuine load-bearing gap. Interestingly, Example 1 does not demonstrate a failure of the claim because the computed K cancels the sin(sin 2t) term in the steady-state residual; the gap is in the theorem statement, not in that specific example. The paper is likely fixable by adding a verifiable condition on Ξ (or enlarging the internal model) and by tightening the assumptions on S. The contribution relative to [6] and [16] is incremental but potentially useful; whether it rises to the level of this journal depends on how convincingly the gaps are closed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the passivation LMI part is a genuine contribution and the algebra is right, but Theorem 2's convergence proof assumes a steady-state trajectory that Assumption 2 does not guarantee. The gap is real but fixable.\n\nWhat is actually new: data-dependent LMIs (20) that enforce incremental passivity via state feedback, and a regulator whose internal-model design is decoupled from the passivation step. That decoupling is an improvement over the earlier data-driven harmonic regulation in [17], and the data representation from [16] is used cleanly. Lemma 5 and Theorem 1 are sound; I checked the matrix manipulations.\n\nThe soft spot: Theorem 2 asserts that system (24) admits a bounded solution ζss = (x(w), ηss) with e=0. For that, ηss must satisfy ηdot = Sη and Ξ^T ηss = u(w) − KZ(x(w)). Assumption 2 guarantees only the plant-level pair (x(w), u(w)); it says nothing about whether the residual lies in the output space of the internal model. No assumption on Ξ or design rule is stated. This is load-bearing because the whole Barbalat argument compares against ζss. Without it, the convergence proof is unsupported.\n\nThe stress-test note says Example 1 demonstrates the gap. It doesn't: in that example the designed K3 = 1 cancels the sin(sin 2t) term in the residual, so the residual has only frequencies 0 and 2 and the required ηss exists. The general gap remains regardless. A fix is to add an explicit condition—e.g., that u(w) − KZ(x(w)) belongs to the span of Ξ^T over the S-invariant subspace—or give a constructive rule for choosing Ξ (and possibly exploiting freedom in K).\n\nA smaller issue: the internal model's passivity with the simple quadratic storage requires S + S^T ≤ 0. The paper's rearrangement of S into the block-diagonal form in Section 2 implicitly assumes S is diagonalizable, which is fine for bounded exosystem signals with a compact invariant set, but it is not stated. Also, no code or data is shipped, so the examples are hard to independently reproduce.\n\nBottom line: the task this paper attempts is worth solving and the first half is solid. The regulation theorem needs another pass, but this is referee-fixable, not a kernel of an idea that should be rejected.\n\nRecommendation: send it to peer review, and ask the authors to repair the ζss existence condition and state the S structural assumption.","headline":"Solid data-driven passivation LMIs with a decoupled internal model, but Theorem 2's convergence proof assumes a steady-state trajectory that Assumption 2 does not guarantee; fixable and worth refereeing.","tokens_in":17717,"tokens_out":12122,"would_cite":false,"duration_ms":109307,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B52","93C10","93D15","93D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Nonlinear output regulation can be solved from offline data: the paper proves that a data-driven passivating gain plus an internal model built from the known exosystem keeps all closed-loop trajectories bounded and drives the regulation…","keywords":["data-driven control","output regulation","nonlinear systems","incremental passivity","passivity-based control","linear matrix inequalities","internal model","trajectory tracking"],"falsifier":"Find a plant that satisfies Assumption 2 (so $x_{\\mathrm{ss}}(w)$ and $u_{\\mathrm{ss}}(w)$ are polynomials in $w$) but whose feature-map evaluation $Z(x_{\\mathrm{ss}})$ contains frequencies outside the exosystem spectrum, and check numerically whether a bounded $\\eta_{\\mathrm{ss}}$ with $\\dot\\eta_{\\mathrm{ss}}=S\\eta_{\\mathrm{ss}}$ and $\\Xi^\\top\\eta_{\\mathrm{ss}}=u_{\\mathrm{ss}}-KZ(x_{\\mathrm{ss}})$ exists for the passivating gain $K$ produced by the LMIs; if no such $\\eta_{\\mathrm{ss}}$ exists, the comparison trajectory used in the proof is unavailable and the theorem's conclusion does not follow for that allowed instance.","tokens_in":16717,"feed_emoji":"🎯","tokens_out":16018,"duration_ms":135176,"temperature":0.7,"pith_summary":"This paper claims that for a class of nonlinear systems with unknown coefficient matrices, the output-regulation problem—asymptotic reference tracking and disturbance rejection—can be solved from a finite set of offline data plus knowledge of the exosystem that generates references and disturbances. The regulator combines a static data-driven feedback gain, characterized by linear matrix inequalities on the measured data that enforce incremental passivity, with an internal model constructed directly from the known exosystem. The paper's central result is that whenever the data-dependent LMIs are feasible and the standard regulator-equations assumptions hold, all closed-loop solutions stay bounded and the regulation error converges to zero. The same machinery also stabilizes a non-zero equilibrium when the equilibrium input is unknown. This matters because it offers a model-free route to asymptotic tracking in which the passivation step and the internal-model step are decoupled.","feed_headline":"Offline data can yield exact tracking for unknown nonlinear systems","feed_subtitle":"A passivity-based regulator built from offline LMIs drives the regulation error to zero without identifying the system matrices.","key_machinery":"The load-bearing mechanism is the data-based closed-loop representation combined with incremental passivity as the design target. From the offline samples one builds $X_0,Z_0,X_1,U_0,E_0$ and the exosystem-derived matrix $M_0$; any $G=(G_1,G_2)$ satisfying (17) produces data-based matrices $A_d=X_1G_1$, $B_d=X_1G_2$, $C_d=E_0G_1$ that agree with the true closed loop on the data. The passivity conditions of Lemma 3---$I^\\top P(A+BK)+(\\cdot)^\\top \\preceq 0$ and $I^\\top PB=C^\\top$---are then rewritten as LMIs (20) in variables $Y$, $G_2$, and a block-diagonal $P=\\mathrm{blockdiag}(P_1,P_2)$, with storage function $V(x_1,x_2)=\\frac12(x_1-x_2)^\\top P_1^{-1}(x_1-x_2)$. The gain $K=U_0YP^{-1}$ is fixed by data alone, while the internal model $\\dot\\eta=S\\eta-\\alpha\\Xi e$ is chosen from the known exosystem, so the two designs are independent; the interconnection lemma for incrementally passive systems supplies the augmented storage function that makes the convergence argument work.","core_discovery":"On the paper's own terms, the discovery is that incremental passivity can be enforced directly from data and then used as the organizing principle for output regulation. The plant is written as $\\dot x = A Z(x) + B u + E w$, $e = C Z(x) + F w$ with unknown $A,B,C,E,F$ and a known feature map $Z(x)$; Lemma 5 shows that the closed-loop matrices $A+BK$, $B$, $C$ can be represented by the data matrices $X_1 G_1$, $X_1 G_2$, $E_0 G_1$ whenever the compatibility equation (17) is solvable. This turns the model-based passivation conditions of Lemma 3 into the data-dependent LMIs (20), whose feasibility is the design certificate. Theorem 2 then states that with those LMIs feasible, the regulator $\\dot\\eta = S\\eta - \\alpha\\Xi e$, $u = KZ(x) + \\Xi^\\top\\eta - \\hat K e$ renders the augmented system incrementally passive with respect to a steady-state comparison trajectory, and Barbalat's lemma converts the resulting dissipation inequality into $\\lim_{t\\to\\infty} e(t)=0$ with bounded closed-loop trajectories. The method also yields, via a virtual output $e_v = \\|x-x_e\\|^2$, global asymptotic stabilization at a non-zero equilibrium without estimating the equilibrium input.","pith_inferences":["Beyond the paper: the decoupling of passivation from the internal model suggests a modular control architecture in which a single data-driven passivating gain is precomputed per plant and stored, and only the internal model is swapped when the reference or disturbance task changes.","Beyond the paper: the proof's comparison trajectory requires the internal model to generate the steady-state input, a condition beyond Assumption 2; a practical safeguard is to check whether $\\Xi^\\top\\eta_{\\mathrm{ss}}=u_{\\mathrm{ss}}-KZ(x_{\\mathrm{ss}})$ is solvable for a bounded $\\eta_{\\mathrm{ss}}$, or to enlarge the internal model with additional harmonics when $Z(x_{\\mathrm{ss}})$ contains fr","Beyond the paper: the virtual-output reformulation suggests the same machinery can handle objectives beyond equilibria---any regulation task expressible as a known polynomial error $e_v=C_vZ_v(x)+F_vw_v$ could be attacked with the same data-driven passivation."],"forward_implications":["If Theorem 2 is right, asymptotic tracking and disturbance rejection for this class of nonlinear systems need no identification of $A,B,C,E,F$: feasibility of the data-dependent LMIs (20) is a direct design certificate.","Because the passivating gain is independent of the internal model, changing the reference or disturbance spectrum only requires replacing the internal model (and recollecting data to reflect the new exosystem), not redesigning the passivation controller.","The same data-driven passivation, with the virtual output $e_v=\\|x-x_e\\|^2$, yields global asymptotic stabilization at a non-zero equilibrium without knowing or estimating the equilibrium input, while rejecting time-varying disturbances.","For linear plants ($Z(x)=x$), the result reduces to Corollary 1, a data-driven linear output regulator with the same passivity-based proof structure.","The design goes beyond earlier data-driven nonlinear regulation of periodic references, which gave bounded tracking error, to asymptotic error convergence for step-plus-sinusoidal exosystems."],"supporting_citations":[{"why":"Defines incremental passivity, regular storage functions, and the interconnection lemma used to combine the closed-loop system with the internal model.","marker":"[6]"},{"why":"Provides the data-based trajectory representation and the Lipschitz condition that the paper adapts to enforce passivity from offline data.","marker":"[16]"},{"why":"The closest prior data-driven nonlinear output regulation result, contrasted here as yielding bounded rather than asymptotic tracking.","marker":"[17]"},{"why":"Establishes informativity conditions for the linear data-driven algebraic regulator problem that this nonlinear formulation extends.","marker":"[14]"},{"why":"Supplies Barbalat's lemma, the final step converting the dissipation inequality and boundedness into asymptotic error convergence.","marker":"[28]"}],"fun_headline_variants":["Data-enforced passivity achieves exact tracking without system ID","Offline data enables a regulator for exact nonlinear tracking","Passivity from data alone solves output regulation for nonlinear systems","Data-driven regulator drives error to zero with no model identification","Incremental passivity from data yields exact tracking for unknown plants"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the internal model can reproduce the steady-state part of the control input---the proof needs a bounded $\\eta_{\\mathrm{ss}}$ following $\\dot\\eta=S\\eta$ with $\\Xi^\\top\\eta_{\\mathrm{ss}}=u_{\\mathrm{ss}}-KZ(x_{\\mathrm{ss}})$---while Assumption 2 only guarantees a plant-level steady state $(x_{\\mathrm{ss}},u_{\\mathrm{ss}})$ exists, and the paper gives no rule for choosing $\\Xi$ to make the matching possible.","fun_headline_variants_meta":{"raw":{"variants":["Data-enforced passivity achieves exact tracking without system ID","Offline data enables a regulator for exact nonlinear tracking","Passivity from data alone solves output regulation for nonlinear systems","Data-driven regulator drives error to zero with no model identification","Incremental passivity from data yields exact tracking for unknown plants"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000393,"raw_usage":{"total_tokens":2076,"prompt_tokens":971,"completion_tokens":1105,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":587,"completion_tokens_details":{"reasoning_tokens":1023}},"tokens_in":587,"tokens_out":1105,"duration_ms":8163,"temperature":1.0,"reasoning_tokens":1023,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T06:03:32.106341+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a plant that satisfies Assumption 2 (so $x_{\\mathrm{ss}}(w)$ and $u_{\\mathrm{ss}}(w)$ are polynomials in $w$) but whose feature-map evaluation $Z(x_{\\mathrm{ss}})$ contains frequencies outside the exosystem spectrum, and check numerically whether a bounded $\\eta_{\\mathrm{ss}}$ with $\\dot\\eta_{\\mathrm{ss}}=S\\eta_{\\mathrm{ss}}$ and $\\Xi^\\top\\eta_{\\mathrm{ss}}=u_{\\mathrm{ss}}-KZ(x_{\\mathrm{ss}})$ exists for the passivating gain $K$ produced by the LMIs; if no such $\\eta_{\\mathrm{ss}}$ exists, the comparison trajectory used in the proof is unavailable and the theorem's conclusion does not follow for that allowed instance.","supporting_citations":[{"cited_title":"Incremental passivity and output regulation,","cited_arxiv_id":null,"evidence_quote":"Defines incremental passivity, regular storage functions, and the interconnection lemma used to combine the closed-loop system with the internal model."},{"cited_title":"Enforcing contraction via data","cited_arxiv_id":"2401.07819","evidence_quote":"Provides the data-based trajectory representation and the Lipschitz condition that the paper adapts to enforce passivity from offline data."},{"cited_title":"Data-driven harmonic output regulation of a class of nonlinear systems,","cited_arxiv_id":null,"evidence_quote":"The closest prior data-driven nonlinear output regulation result, contrasted here as yielding bounded rather than asymptotic tracking."},{"cited_title":"An informativity approach to the data-driven algebraic regulator problem,","cited_arxiv_id":null,"evidence_quote":"Establishes informativity conditions for the linear data-driven algebraic regulator problem that this nonlinear formulation extends."}],"review_version":1}