{"id":"d8eba424-8a86-401c-a8b3-e54f28456e37","arxiv_id":"2505.19131","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":7,"one_line_summary":"A two-controller architecture that uses a funnel controller to guarantee output constraints while a DeePC or EDMD-based predictive controller learns, permitting safe online data collection and tracking.","lead":"This paper combines a learning-based predictive controller with a model-free funnel controller, so that output constraints are always enforced even when the learned model is poor. The design is demonstrated for reference tracking on a Van der Pol oscillator, with the safeguarding controller also used to collect training data.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Safety guarantee is conditional on an unproven, unenforced boundedness of the predictive input and on a proof sketch of Theorem 1 that may not cover the activation-controlled funnel; the paper's central safe-learning claim is not yet established.","rationale":"The reader's conditional verdict is appropriate. The weakest assumption identified by the reader, boundedness of µ, is also the central gap in the paper's safety claim: Theorem 1 assumes it rather than deriving it, and the controller architecture provides no mechanism guaranteeing feasibility or boundedness of the predictive input. The proof of Theorem 1 is only sketched and the citation to prior work does not transparently cover the activation-scaled funnel controller with potentially small aτ in the critical zone; this is a concrete technical point that could invalidate the theorem. The paper is honest about the unfinished fill-distance scheme, so that portion cannot support the approximation-accuracy claims. However, the core idea is plausible, the numerical examples illustrate the intended behaviour, and no outright contradiction was found in the parts that are actually proven. Thus the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":18981,"tokens_out":6719,"duration_ms":72699,"concrete_test":"Two-part check. (1) Formal/numerical falsification of Theorem 1: simulate the closed loop (1),(4),(5),(6) for the Van der Pol system (15) with the parameters of Section 5, but replace the EDMD-MPC by worst-case bounded inputs µ(t)≡B for B∈{1,2,5,10}, sweeping λ∈{0.5,0.75,0.9} and initial conditions satisfying ∥e1(0)∥,∥e2(0)∥<0.9; if constraint (3) is ever violated, Theorem 1 is false as stated. (2) Feasibility certificate: instrument Algorithm 1 to record all OCP infeasibility events during the run of Section 5.2.2 and exhibit the fallback value of µ on those occasions; verify that with this fallback the closed loop still satisfies (3).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central safety guarantee, Theorem 1, is not actually proven here and rests on an assumption that is not enforced by the proposed controller. The proof is a sketch that invokes [31, Thm. 1] and [10, Thm. 1.9], but the present closed loop differs from those settings: the activation function (6) scales uFC by aτ = max{0, max_{s∈[t−τ,t]}∥e2(s)∥−λ}, which is arbitrarily small when the error first enters the critical zone ∥e2∥∈[λ,1). The domination argument used in standard funnel control assumes the full feedback gain; the sketch gives no lower bound on aτ in the critical zone, so it is not established that the scaled feedback can dominate a bounded but otherwise arbitrary µ. Moreover, the theorem explicitly assumes µ is bounded, but Algorithm 1 only outputs a control if the finite-horizon OCP is feasible, and feasibility is neither certified nor replaced by a fallback value; a poor EDMD surrogate can make the OCP infeasible or produce µ outside any a priori bound. The same gap applies to the DeePC variant (8), whose feasible set depends on Hankel data. Hence the claim 'safe learning without offline training' is conditional on an unverified property of the learning component. The fill-distance data-collection scheme (Fig. 3) is explicitly deferred to future research, so it cannot support the approximation-accuracy claims either.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a two-component feedback architecture for output tracking of control-affine systems of relative degree two: a learning-based predictive controller (exemplified by DeePC and by EDMD-based MPC) provides the nominal input, and a model-free funnel controller is added through an activation function that switches it on only when a scaled error variable leaves a safe λ-region. Theorem 1 claims that, whenever the predictive input is bounded, the closed loop satisfies the prescribed error bound ||y(t)-y_ref(t)|| < 1/σ(t) for all time. The paper further sketches a fill-distance-based data collection procedure intended to guarantee a desired approximation accuracy for kernel-EDMD surrogates, and illustrates the approach numerically on a forced Van der Pol oscillator.","tokens_in":19335,"tokens_out":8015,"duration_ms":76357,"significance":"If the safeguarding property were fully proven and the boundedness/feasibility of µ were enforced, the architecture would be a useful step toward safe learning-based predictive control without offline training: it combines two established components in a modular way, and the explicit link to fill-distance guarantees for kernel EDMD is original. The paper is clearly written and the numerical examples illustrate the behavior. However, the main theoretical claim is only sketched and relies on assumptions not certified by the algorithms, so the central 'safe learning' guarantee is not yet established in the manuscript. No machine-checked proofs, parameter-free derivations, or reproducibility artifacts are provided; the numerical results are illustrative only.","major_comments":[{"comment":"The proof of Theorem 1 is not self-contained: it invokes [31, Thm. 1], [10, Thm. 1.9] and [46, Thm. 5.1], none of which includes the activation factor aτ defined in (6). In the critical zone ||e2(t)|| ∈ [λ, 1), aτ(t, e2) = max{0, max_{s∈[t−τ,t]} ||e2(s)|| − λ} can be arbitrarily small: for an error that has just entered the zone, the most recent maximum is close to ||e2(t)||, so aτ is close to zero at the boundary. The standard funnel-control proof dominates a bounded disturbance by the full gain −e2/(1−||e2||^2); with the gain multiplied by a vanishing factor, that domination argument does not go through. The claim that there exists ε̃ ∈ [λ, 1) with ||ei(t)|| ≤ ε̃ for i = 1, 2 requires a positive lower bound on aτ during the activation intervals, which the manuscript does not provide. This gap is load-bearing because all subsequent safe-learning statements rest on Theorem 1. If the result is already proven under these conditions in the authors' earlier work, the manuscript should state that explicitly and give the precise theorem mapping; as written, the proof is only a sketch.","section":"Section 3, Theorem 1 and proof sketch"},{"comment":"Theorem 1 assumes that the predictive input µ is bounded, but neither controller component enforces this. Algorithm 1 solves a finite-horizon OCP in Step 2 without a feasibility certificate; if the EDMD surrogate is poor, the OCP may be infeasible, in which case no µ is produced and control law (4) is undefined. The DeePC formulation (8) imposes the input bound ||u(i)|| ≤ umax, but it does not prove that the optimization is feasible at every step, nor that the resulting µ satisfies the boundedness hypothesis of Theorem 1. The numerical example imposes U = [−2, 2] in Algorithm 1, but this is an implementation choice not covered by the theorem's assumptions. To substantiate the central claim, the authors should either add an explicit fallback (e.g., µ = 0 whenever the OCP is infeasible) and include it in Theorem 1, or prove recursive feasibility and an input bound for the chosen predictive component.","section":"Section 2.2, Algorithm 1, and Eq. (8)"},{"comment":"The fill-distance data-collection scheme is not a theorem. The crucial step 'Define reference yref and sampling time Δt such that S_{i=0}^D {(yref(iΔt), yrefdot(iΔt))} = X' is an assumption on the design, not a construction, and the manuscript itself states in Section 6 that the scheme 'will be thoroughly analyzed' in future research. Equation (14) bounds ||x(t) − (yref(t), yrefdot(t))|| by 3/σ; to conclude x(iΔt) ∈ B_{εc}(x_i), one must also know that the reference evaluated at iΔt equals x_i, which is imposed but not established for a function yref ∈ W^{2,∞}. Thus the statement in Fig. 3 that the EDMD-based surrogate satisfies the bounds of Theorem 3 is premature. The authors should either provide the construction/proof or clearly mark this scheme as a conjecture/outlook rather than part of the established contributions.","section":"Section 4.2.3, Fig. 3"}],"minor_comments":[{"comment":"The proof sketch refers to 'multiplying uFC by ατ', but the notation used elsewhere, including Eq. (6), is aτ; the notation should be made consistent.","section":"Section 3, proof sketch"},{"comment":"The set notation 'Sd_i=1' in Fig. 3 should be replaced by a union symbol or explicitly defined, and the parameter D used in the text below the figure should be defined in the figure caption.","section":"Fig. 3"},{"comment":"The reference y_t_ref(t) is defined with a parameter t̂ that is not formally introduced, and the symbols y10_ref and y16_ref in Sections 5.2.1 and 5.2.2 are used without explicit formulas; please clarify their definitions.","section":"Section 5.2"},{"comment":"When citing [46, Lem. 2.1] for the bounds on ||e1(t)|| and ||e2(t)||, the text should state explicitly that this lemma concerns the pure funnel controller (5), not the two-component controller (4), to avoid ambiguity.","section":"Section 4.2.3"},{"comment":"The caption 'Set-point transition ,initializing with 1 data point' contains a spacing typo and should read 'Set-point transition, initializing with 1 data point'.","section":"Caption of Fig. 9"}],"recommendation":"major_revision","confidential_remarks":"The central theorem's proof is deferred to earlier papers by the same group, and several approximation-theoretic ingredients are taken from the authors' own preprints. This is not by itself a flaw, but the editor may wish to weigh whether the tutorial-style presentation with a sketched main proof and an explicitly deferred data-collection scheme provides sufficient new technical content for a research journal. The modular two-component idea is potentially valuable; the revision should either supply a complete proof of Theorem 1 and a certified fallback for µ, or state clearly which parts remain conjectural."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a tutorial-style paper that puts funnel control as a safety layer around data-driven predictive control, with DeePC and EDMD-MPC as the two concrete instantiations. The genuinely new pieces are small: the EDMD-MPC version of the two-component idea, and a sketched procedure for steering the system through virtual observation points to control the fill distance. The paper is upfront that the second piece is unfinished.\n\nWhat it does well: the architecture is explained clearly, the activation function with dwell-time is a sensible anti-chattering mechanism, and the Van der Pol example shows the idea works in practice. The paper is honest about what is future work, and the heavy self-citation is to results that are genuinely the right ones to cite.\n\nSoft spots: Theorem 1 is load-bearing but its proof is a sketch. The step where “the analysis in [10, Thm. 1.9] applies” after multiplying the funnel feedback by aτ is not automatic. In the critical zone the activation can be arbitrarily small when e2 first crosses λ, and standard funnel control proofs use the full feedback gain. The argument can likely be patched because near |e2|=1 the activation is bounded below by 1−λ, but the paper should show it. Second, Theorem 1 assumes µ is bounded, but Algorithm 1 has no fallback if the OCP is infeasible. Setting µ=0 on infeasibility is an easy fix, but it should be stated. Third, the fill-distance scheme is explicitly future work, so it cannot yet support the approximation-accuracy claims.\n\nWho this is for: readers wanting a quick, readable introduction to safe online learning with funnel control, or a starting point for combining data-driven predictive control with a model-free safety net. It is not a self-contained research paper.\n\nRecommendation: worth a serious referee if the venue accepts tutorial content. A referee should ask for a complete proof of Theorem 1, with the adaptation to the activation function spelled out, and a brief discussion of feasibility handling. With those, it would be a solid tutorial.","headline":"A tutorial-style paper that applies the authors' funnel-safeguard architecture to EDMD-MPC and DeePC; the fill-distance data-collection idea is genuinely new but explicitly unfinished, and Theorem 1's proof is a sketch that needs tightening.","tokens_in":19823,"tokens_out":5362,"would_cite":false,"duration_ms":41440,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C10","93B52","93C57","93D15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For output tracking on second-order control-affine systems, a model-free funnel controller guarantees the prescribed error bound even while a data-driven predictive controller is learning, provided the predictive input stays bounded.","keywords":["two-component controller","funnel control","data-driven predictive control","data-enabled predictive control (DeePC)","EDMD-based MPC","Koopman operator","fill distance","safe learning"],"falsifier":"Run the set-point transition of the forced oscillator example of Section 5 with an EDMD dictionary chosen to be deliberately poor, for instance one that omits the coordinate functions or uses a kernel whose fill distance exceeds the threshold of Theorem 3; if the computed input $\\mu$ grows without bound and the closed-loop output leaves the prescribed funnel while Assumptions 1 and 2 hold and the initial auxiliary errors are inside the unit ball, then the boundedness premise of Theorem 1 is violated in practice.","tokens_in":18761,"feed_emoji":"🛡️","tokens_out":16135,"duration_ms":89152,"temperature":0.7,"pith_summary":"The paper proposes a two-component controller for output tracking with prescribed error bounds on a class of second-order control-affine systems, with no model of the plant required. One component is a data-driven predictive controller, exemplified by data-enabled predictive control (DeePC) and by model predictive control built on extended dynamic mode decomposition (EDMD); the other is a model-free funnel controller that an event-triggered activation function switches on only when the tracking error leaves a safe region. The central claim, Theorem 1, is that this combination keeps the tracking error strictly inside the prescribed funnel for all time, no matter how poor the learned model is, as long as the predictive input is bounded and the initial auxiliary errors start inside the funnel. The same funnel mechanism doubles as a data collector: steering along a reference that visits prescribed virtual observation points keeps samples within a chosen cluster radius, so the fill distance needed for kernel-EDMD approximation error bounds can be achieved. This is a route to safe learning-based predictive control without an offline training phase.","feed_headline":"A safety loop guarantees tracking while data-driven controllers learn","feed_subtitle":"A model-free funnel controller backs up learned models, so error bounds hold even with one data point.","key_machinery":"The load-bearing mechanism is the two-component feedback law $u = \\mu(t,x) + a_\\tau(t,e_2) u_{\\mathrm{FC}}(t)$, with the model-free funnel law $u_{\\mathrm{FC}} = -e_2/(1 - \\|e_2\\|^2)$ built from the scaled error variables $e_1 = \\sigma(t)(y - y_{\\mathrm{ref}})$ and $e_2 = \\sigma(t)\\dot{e} + e_1/(1 - \\|e_1\\|^2)$. The activation function $a_\\tau = \\max\\{0, \\max_{s \\in [t-\\tau,t]} \\|e_2(s)\\| - \\lambda\\}$, with threshold $\\lambda \\in (0,1)$ and dwell time $\\tau > 0$, keeps the safeguard switched off in the safe region and prevents chattering when it switches on. The funnel feedback is high-gain: as $\\|e_2\\|$ approaches 1 the denominator vanishes, which forces the auxiliary variables back into the unit ball and hence keeps $\\|y(t) - y_{\\mathrm{ref}}(t)\\| < 1/\\sigma(t)$. For the EDMD instance, the second key object is the bilinear surrogate $x^+ = [I_n, 0] K_u^{\\Delta t} \\Psi(x)$, formed by lifting states into observables and approximating the Koopman operator from data; the fill distance $h_\\mathcal{X}$ and cluster radius $\\varepsilon_c$ of the collected samples certify the surrogate's approximation error.","core_discovery":"For systems of the form $\\dot{x}_1 = x_2$, $\\dot{x}_2 = g_0(x) + G(x)u$, $y = x_1$ with sign-definite input distribution $G$, the paper claims that the two-component feedback $u = \\mu + a_\\tau u_{\\mathrm{FC}}$, where $\\mu$ is any bounded input produced by a data-driven predictive controller and $u_{\\mathrm{FC}}$ is the funnel feedback, guarantees $\\|y(t) - y_{\\mathrm{ref}}(t)\\| < 1/\\sigma(t)$ for all $t \\geq 0$ whenever the funnel function and reference are feasible and the auxiliary variables $e_1, e_2$ start inside the unit ball. The proof is given only as a sketch and deferred to prior work: outside the safe $\\lambda$-region the funnel feedback acts with increasing gain as $\\|e_2\\|$ approaches 1, driving the auxiliary variables back inside, while inside the safe region the activation function keeps the funnel inactive. The paper also claims that this safeguarding property supports safe online learning: EDMD-based MPC can start with a single data point and still track a set-point transition within the prescribed error bounds, with the funnel controller active only briefly during the transient. Finally, the same feedback mechanism is used to steer the system so that samples land near prescribed virtual observation points, making the fill-distance-dependent kernel-EDMD approximation error bound applicable.","pith_inferences":["A natural hardening step would be to make the boundedness of $\\mu$ a certified property, for example by adding hard input constraints and proving feasibility of the MPC problem under the surrogate's error bounds; without that, the theorem's main assumption is left to the optimizer's good behavior.","The fill-distance argument turns data collection into a planning problem: choose virtual observation points and a reference so that a target approximation error is reached with a minimal number of samples, and use the activation function's record of interventions as a data-quality flag.","The safety argument is likely not tied to funnel control specifically: prescribed performance control and other high-gain safeguards for the same system class should combine with the same activation function, so the architecture is a template for safe learning rather than a single controller.","A quantitative trade-off worth testing is between exploration speed and data quality: larger $\\sigma$ shrinks the neighborhood visited around the reference, making fill-distance control easier but slowing the coverage of the state-space region."],"forward_implications":["For any data-driven predictive controller whose input is bounded, output constraint satisfaction holds during runtime, so the safeguarded schemes need no offline training phase.","Steering the system along a reference through prescribed virtual observation points with $\\sigma \\geq 3/\\varepsilon_c$ keeps sampled states within $\\varepsilon_c$ of those points, so the fill-distance-dependent error bounds of Theorem 3 apply to the learned EDMD surrogate.","The EDMD-based MPC can be initialized with a single data point and still complete a set-point transition inside the prescribed error bounds, with the funnel controller activating only a few times.","DeePC combined with the funnel controller inherits the same safeguarding guarantee, and the data needed for its Hankel matrices can be collected online rather than in advance.","Because the funnel component is model-free and independent of the prediction scheme, the two-component architecture applies to any learning-based predictive controller, not just the two examples worked out here."],"supporting_citations":[{"why":"Supplies the funnel-control feasibility theorem that the safeguarding proof's sketch invokes (in particular [10, Thm. 1.9]).","marker":"[10]"},{"why":"Introduces data-enabled predictive control (DeePC), the first data-driven predictive instance used in the two-component design.","marker":"[21]"},{"why":"Provides the activation-function design and the invariance reasoning that Theorem 1's proof sketch follows.","marker":"[31]"},{"why":"Supplies the sampled-data funnel-control analysis and the lemma giving the state-error estimate used to choose $\\sigma$ for fill-distance control.","marker":"[46]"},{"why":"States the fundamental lemma that justifies the trajectory parameterization underlying the DeePC optimization problem.","marker":"[77]"},{"why":"Shows the combination of funnel control with DeePC and discusses adapting the prediction horizon from available data.","marker":"[68]"},{"why":"Establishes practical asymptotic stability of EDMD-based MPC under finite-data error bounds, the stability result the safeguarded scheme inherits.","marker":"[13]"},{"why":"Provides finite-data approximation error bounds for control-affine kernel-based surrogates, including the zero-order-hold discretization error.","marker":"[14]"},{"why":"Gives the fill-distance and cluster-radius error bound (Theorem 3) that the data-collection scheme is designed to satisfy.","marker":"[16]"},{"why":"Supplies the $L^\\infty$-error bound for kernel EDMD on autonomous systems that motivates the fill-distance sampling guarantee.","marker":"[40]"}],"fun_headline_variants":["Safety funnel backs data-driven predictive control","Two-component controller: learn and safeguard in parallel","Data-driven MPC with a model-free safety net","Guaranteed tracking even with a single data point","Hybrid control: predictive learning plus funnel safety"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem depends on the predictive controller's input being bounded, an assumption the paper states rather than derives from the learned surrogate and the optimization that computes the input, and it also presumes the reference trajectory can be routed through the prescribed virtual observation points at sampling instants.","fun_headline_variants_meta":{"raw":{"variants":["Safety funnel backs data-driven predictive control","Two-component controller: learn and safeguard in parallel","Data-driven MPC with a model-free safety net","Guaranteed tracking even with a single data point","Hybrid control: predictive learning plus funnel safety"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000258,"raw_usage":{"total_tokens":1624,"prompt_tokens":1027,"completion_tokens":597,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":643,"completion_tokens_details":{"reasoning_tokens":527}},"tokens_in":643,"tokens_out":597,"duration_ms":6230,"temperature":1.0,"reasoning_tokens":527,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:19:15.518879+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the set-point transition of the forced oscillator example of Section 5 with an EDMD dictionary chosen to be deliberately poor, for instance one that omits the coordinate functions or uses a kernel whose fill distance exceeds the threshold of Theorem 3; if the computed input $\\mu$ grows without bound and the closed-loop output leaves the prescribed funnel while Assumptions 1 and 2 hold and the initial auxiliary errors are inside the unit ball, then the boundedness premise of Theorem 1 is violated in practice.","supporting_citations":[{"cited_title":"Funnel control of nonlinear systems","cited_arxiv_id":null,"evidence_quote":"Supplies the funnel-control feasibility theorem that the safeguarding proof's sketch invokes (in particular [10, Thm. 1.9])."},{"cited_title":"Data-enabled predictive control: In the shallows of the DeePC","cited_arxiv_id":null,"evidence_quote":"Introduces data-enabled predictive control (DeePC), the first data-driven predictive instance used in the two-component design."},{"cited_title":"Reinforce- ment Learning for Docking Maneuvers with Pre- scribed Performance","cited_arxiv_id":null,"evidence_quote":"Provides the activation-function design and the invariance reasoning that Theorem 1's proof sketch follows."},{"cited_title":"Sampled-data fun- nel control and its use for safe continual learning","cited_arxiv_id":null,"evidence_quote":"Supplies the sampled-data funnel-control analysis and the lemma giving the state-error estimate used to choose $\\sigma$ for fill-distance control."},{"cited_title":"A note on persistency of excitation","cited_arxiv_id":null,"evidence_quote":"States the fundamental lemma that justifies the trajectory parameterization underlying the DeePC optimization problem."},{"cited_title":"Safe data-driven reference tracking with prescribed performance","cited_arxiv_id":null,"evidence_quote":"Shows the combination of funnel control with DeePC and discusses adapting the prediction horizon from available data."},{"cited_title":"Data-driven MPC with stability guarantees using extended dynamic mode decom- position","cited_arxiv_id":null,"evidence_quote":"Establishes practical asymptotic stability of EDMD-based MPC under finite-data error bounds, the stability result the safeguarded scheme inherits."}],"review_version":1}