{"id":"3f635b01-6c58-49e0-b32d-1af96effb700","arxiv_id":"2507.02549","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"A Koopman-based adaptive backstepping controller is claimed to achieve prescribed-time stabilization for nonlinear systems, but the proof is incomplete and the control law contains an undefined term.","lead":"This paper proposes a controller that combines a data-learned Koopman linear model with adaptive backstepping to stabilize certain nonlinear systems within a user-chosen time. The method is validated only with a Van der Pol simulation, and the derivation leaves key terms undefined, so the central claim is not yet established.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Load-bearing flaw: virtual controls contain the actual input u through (Bu)_i, so Eq. (12a) is an implicit equation, not a well-defined controller; no sparsity assumption on B is stated.","rationale":"The reader's weakest_assumption identifies the issue I find most load-bearing: the proposed control law is not actually a function of the state. The reason is not a numerical detail but a structural failure of the recursive construction. In standard backstepping, the input appears only in the last virtual-coordinate equation, so alpha_i are independent of u. Here the Koopman-lifted model is allowed to be fully dense, so u enters every dot z_i through (Bu)_i. The paper explicitly claims this generality, but that generality is what makes the design circular. Either the authors must add and justify a sparsity or dependency assumption on B, or they must solve the implicit equation for u and present the solution. Without this, Eqs. (9)-(12) do not define a controller, and the subsequent Lyapunov inequality cannot be evaluated. The undefined Phi_N(z) is a second, independent blocker, but the algebraic/implicit loop is the more fundamental concern because it bears on the existence of the feedback law itself. My concrete test, carrying out the elimination for N=2, would decisively show the loop; if the authors can exhibit an explicit solution or an added assumption that rules out the offending B entries, the concern would be resolved. As submitted, the central claim is not supported by the stated construction. I therefore find no reason to alter the reader's REJECT verdict.","tokens_in":8184,"tokens_out":4368,"duration_ms":53600,"concrete_test":"Specialize to N=2 with z_dot = A z + B u, B = [b1; b2], b1 nonzero. Write alpha_1 from Eq. (9): alpha_1 = -c1 rho e1 - (Az)_1 - b1 u + dot z_{d,1}. Substitute this into the final control expression (12a), keeping the Phi_N term symbolic, and differentiate alpha_1: dot alpha_1 contains -b1 dot u. The resulting equation for u contains dot u, so it is not a static state-feedback map. Then check whether any condition in Sections II-III implies b1 = 0; the text imposes none, and EDMDc does not. Performing this elimination for general N with dense B exhibits the same circularity. This analytical check settles whether the control law is well-defined: if u cannot be eliminated explicitly without an added sparsity assumption, the recursive design is not a valid feedback law.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central PPTS claim rests on the recursive design being an explicit state-feedback law. In Section III-B, every virtual control alpha_i (Eqs. (9) and (11)) includes -(Bu)_i. For a scalar input, (Bu)_i = B_{i,1} u, so alpha_i depends on the actual control u. The final control law (12a) contains alpha_{N-1} and its derivative; since alpha_{N-1} contains u, dot alpha_{N-1} contains dot u. Thus Eq. (12a) is not an explicit expression for u; it defines u only implicitly, through a differential/algebraic loop. No assumption in Section III restricts B to a strict-feedback sparsity pattern (e.g., B_{i,1}=0 for i<N), and EDMDc generically returns dense matrices. The paper even claims the design 'obviates the need for the learned matrices A and B to have a specific sparse structure,' but that generality is exactly what makes the recursion circular. Consequently, the 'controller' is not well-defined for the learned models the paper targets, and the Lyapunov computation in Eqs. (14)-(15) has no concrete closed-loop vector field to bound. A second, independent gap is that Phi_N(z) in Eqs. (12a)-(12b) is never defined anywhere in the manuscript, so even the intended adaptive term is unspecified. Together these gaps prevent implementation and void the formal PPTS guarantee, not merely its tuning.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a data-driven prescribed-time control framework that combines EDMDc-based Koopman modeling with adaptive backstepping. The authors learn a high-dimensional linear model z = Az + Bu + Δ with a bounded uncertainty assumption, then apply recursive backstepping in the lifted space to design a Prescribed-Time Adaptive Backstepping (PTAB) controller. They claim practical prescribed-time stability (PPTS) for a class of parametric strict-feedback nonlinear systems, with tracking errors converging to a small residual set within a user-prescribed time T, and support the claim with a Lyapunov analysis and a Van der Pol oscillator simulation.","tokens_in":8464,"tokens_out":3556,"duration_ms":43963,"significance":"If the claimed guarantees were valid, the paper would present a useful integration of Koopman-based data-driven modeling with prescribed-time control, addressing the gap between black-box data-driven methods and Lyapunov-based model-based control. The attempt to quantify model uncertainty and incorporate it into a Lyapunov design is a worthwhile direction, and the simulation vignette is illustrative. However, the central controller is not well-defined, the stability proof contains substantial gaps, and the connection between the original nonlinear system and the lifted design is not established. As written, the paper does not provide a valid or implementable control framework, and the advertised PPTS guarantee is not proven.","major_comments":[{"comment":"The virtual control laws contain the actual control input: in Eq. (9), α1 includes −(Bu)1, and in Eq. (11), αi includes −(Bu)i. For a generically learned B, (Bu)i involves the final control u, so every virtual control depends on u. The final control law (12a) then contains α_{N-1} and its derivative, making (12a) an implicit differential/algebraic equation rather than an explicit state-feedback law. No sparsity or structural assumption on B is stated; in fact, Section III-A explicitly claims the design 'obviates the need for the learned matrices A and B to have a specific sparse structure,' but this generality is exactly what makes the recursion circular. The controller is therefore not well-defined for the EDMDc-learned dense models the paper targets, and the subsequent Lyapunov analysis does not apply to a concrete closed-loop system.","section":"Section III-B, Eqs. (9), (11), (12a)"},{"comment":"The vector function Φ_N(z) appears in the final control law (12a) and the adaptive law (12b) but is never defined anywhere in the manuscript. The adaptive term is therefore unspecified, and the controller cannot be implemented from the paper. Moreover, the stability analysis of Section IV implicitly assumes that Φ_N(z) satisfies properties that are never stated, so even the intended adaptive design has no definite object to prove stability for.","section":"Section III-B, Eqs. (12a)-(12b)"},{"comment":"The derivation from Eq. (14) to Eq. (15) is not demonstrated; the text states 'after some algebraic manipulation' without presenting the calculation. More importantly, Eq. (15) of the form V̇ ≤ −2ρ(t)c_min V_e − k_v V + D_0 is claimed to guarantee PPTS, but no comparison argument is given that shows the error enters a residual set by time T independent of initial conditions. The standard UUB conclusion from V̇ ≤ −k_vV + D_0 does not by itself yield a prescribed settling time, and the role of the time-varying term −2ρ(t)c_minV_e in forcing convergence by T is not quantified. Appendix A, which is supposed to provide the needed bound on ||z||, is equally handwavy: its inductive step asserts that α_i, which depends on u implicitly through (Bu)_i, can be bounded by functions of ||e|| and ||θ̃||, but this is exactly what needs proof and is circular given the implicit definition of the controller.","section":"Section IV, Eqs. (14)-(15) and Appendix A"},{"comment":"The paper never establishes the relationship between the original parametric strict-feedback system (1) and the lifted model (5). The unknown parameter θ in (1) does not appear in (5), where all uncertainties are lumped into Δ; consequently, the adaptive law (12b) has no identifiable quantity to estimate. Additionally, the control objective is stated for the original state x, but the backstepping design is carried out in the lifted variable z = Ψ(x), with reconstruction x ≈ Cz used without a rigorous error bound. Without these connections, the framework does not actually solve the stated problem for the class of systems in (1), and the simulation results on the Van der Pol oscillator do not bridge this gap.","section":"Section II-B / III"}],"minor_comments":[{"comment":"The footnote 'This work was supported by... (Please insert your funding information here)' is a template placeholder that should be removed or completed before submission.","section":"Section I, footnote"},{"comment":"The gain function μ(t) is defined in Eq. (7) but is never used later; the design instead uses ρ(t) = 2/(T-t), and the connection between the two is not explained.","section":"Section II-C, Eq. (7)"},{"comment":"The notation (Bu)'_N is introduced only in a parenthetical without a formal definition, which makes the final control law difficult to parse.","section":"Section III-B, Eq. (12a)"},{"comment":"The residual set Ω_e is referred to in the control objective but is never explicitly characterized in terms of the design parameters or uncertainty bounds δ0, δ1.","section":"Section II-D"},{"comment":"The conversion from the discrete-time EDMDc model to the continuous-time model in Eq. (4) is asserted without any details or assumptions on the sampling interval, which is important because Assumption 1 uses the continuous-time residual Δ.","section":"Section II-B"}],"recommendation":"reject","confidential_remarks":"The manuscript has fundamental, load-bearing issues: the control law is implicit, a key function Φ_N(z) is undefined, and the stability proof does not establish the advertised prescribed-time guarantee. These are not presentation issues that could be fixed by a revision; they require a reformulation of the controller and the proof. The simulation section cannot validate a controller that is not fully specified. I also note that the paper contains several template artifacts and a vague 'some algebraic manipulation' step, which suggests the technical content is not yet at the level required for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Name],\n\nThe short version: the paper puts forward a fresh combination of EDMDc-based Koopman modeling and prescribed-time adaptive backstepping, and the authors deserve credit for trying to incorporate a quantified uncertainty bound into the Lyapunov analysis. But the controller as written is not well-defined, and the stability proof is a sketch with the key inequality asserted after 'some algebraic manipulation.' I would reject in its current form, but the idea is worth a serious revision.\n\nWhat is actually new: I don't recall seeing this exact integration in the literature. Lifting a strict-feedback system via EDMDc and then applying backstepping in the lifted space with a prescribed-time gain is a natural but non-obvious step. The paper also explicitly claims that no sparsity structure is needed for A and B, which would be valuable if it worked.\n\nThe soft spots are substantial. In the recursive step, every virtual control α_i contains (Bu)_i. For a generic learned B, that means α_i depends on the actual input u. Consequently the final control law (12a) is an implicit equation—u appears on both sides—and the error variables themselves depend on u. This algebraic loop is never addressed. The paper's assertion that the design obviates the need for a sparse structure is exactly what makes the recursion circular. This is a load-bearing problem, not a stylistic one.\n\nSeparately, Φ_N(z) in (12a)–(12b) is never defined anywhere. Without that, the controller and adaptive law cannot be implemented, and the Lyapunov proof has no concrete object. The appendix's bound on ||z|| is also handwavy: the induction does not track the dependence on the final control, and it uses the controller's boundedness to prove the controller's boundedness.\n\nThe simulation on the Van der Pol oscillator looks encouraging, but figures cannot repair a spec that is missing a key function and an implicit definition. If the authors fill these gaps—define Φ_N, resolve the u-on-both-sides issue (e.g., by restricting B to a lower-triangular structure), and write out the full proof—a revised version could be a legitimate contribution. As it stands, a serious referee should reject but invite resubmission.\n\nI'd bring it to a reading group as an example of how data-driven controller papers can go wrong; I wouldn't cite it yet.\n\nBest,\n[Name]","headline":"A fresh integration of Koopman learning and prescribed-time backstepping, but the controller is not fully defined and the proof is a sketch—reject with a clear path to revision.","tokens_in":8989,"tokens_out":3917,"would_cite":false,"duration_ms":45001,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that an adaptive backstepping controller designed on a Koopman model learned from data achieves practical prescribed-time stability for parametric strict-feedback nonlinear systems, with the tracking error entering a…","keywords":["prescribed-time control","Koopman operator","EDMDc","adaptive backstepping","practical prescribed-time stability","data-driven control","strict-feedback systems","Van der Pol oscillator"],"falsifier":"Write the closed loop down literally: because the virtual controls in (9) and (11) each contain $(Bu)_i$, the control $u$ appears on both sides of its own definition, and the function $\\Phi_N(z)$ in (12a)-(12b) is never defined; a concrete test is to attempt the substitution and see whether an explicit expression for $u$ exists at all.","tokens_in":7932,"feed_emoji":"⏱️","tokens_out":10211,"duration_ms":111564,"temperature":0.7,"pith_summary":"This paper tries to close the gap between model-based control and data-driven control by fitting a linear Koopman model to data with a least-squares method called EDMDc, quantifying how wrong that model is, and then designing an adaptive backstepping controller in the lifted space. The headline claim is that the tracking error enters and stays in a small residual set by any user-chosen time $T$, regardless of initial conditions, while all closed-loop signals remain bounded. This would matter because purely data-driven controllers usually offer no formal stability guarantee, whereas model-based prescribed-time methods require an accurate model. The paper also argues that working in the linear lifted space removes the explosion of complexity that normally forces repeated differentiation of virtual controls in backstepping. A Van der Pol oscillator example with five initial conditions and a sinusoidal disturbance is used to illustrate the claim.","feed_headline":"Koopman plus backstepping claims a user-picked settling time","feed_subtitle":"If the proof holds, nonlinear systems get a pre-assignable convergence time with Lyapunov guarantees.","key_machinery":"The load-bearing objects are the EDMDc-learned linear model $\\dot{z} = Az + Bu + \\Delta$ with error bound $\\|\\Delta\\| \\le \\delta_0 + \\delta_1\\|z\\|$, and the time-varying scaling function $\\rho(t)=2/(T-t)$, equivalently $\\mu(t)=T^2/(T-t)^2$. The scaling function is what turns ordinary Lyapunov convergence into prescribed-time convergence: its divergence near $t=T$ makes the gain dominate any fixed term, so convergence cannot be postponed beyond the user's deadline. The uncertainty bound is what keeps the data-driven step honest: it lumps the Koopman approximation error, state reconstruction error, and external disturbance into one term that the backstepping design absorbs with Young's inequality, leaving a residual set whose size depends on $\\delta_0$ and $\\delta_1$. The recursive construction of virtual controls $\\alpha_i$ in the lifted coordinates is what avoids differentiating the original nonlinearities and hence avoids the explosion of complexity.","core_discovery":"On its own terms, the paper's central discovery is that prescribed-time adaptive backstepping can be carried out entirely in the Koopman-lifted coordinate system $z = \\Psi(x)$, using only a linear model $Az + Bu$ plus a bounded residual $\\Delta$. Under Assumption 1, the recursive design yields the Lyapunov inequality $\\dot{V} \\le -2\\rho(t)c_{\\min}V_e - k_v V + D_0$, where $\\rho(t)=2/(T-t)$ grows without bound as $t$ approaches $T$. The unbounded gain forces the error norm into a residual set before $T$, and the $-k_v V + D_0$ term keeps it bounded afterwards, giving practical prescribed-time stability in the sense of Definition 2. The author treats the structure of the learned linear model as a design asset: because $A$ and $B$ are dense, every $Az$ term is handled as feedforward, so no special sparsity is needed and the derivatives of virtual controls stay simple. That is the mechanism by which data-driven model identification and a Lyapunov-based performance guarantee are claimed to coexist.","pith_inferences":["A natural extension not developed in the paper is to make the dictionary adaptive online, so the uncertainty bounds could be updated as new data arrive and the residual set tightened during operation.","Because the prescribed-time gain diverges as $t$ approaches $T$, any real actuator with finite authority will saturate near the deadline; the guarantee should be read as holding for ideal actuation unless input constraints are added.","The framework's credibility rests on a validation-set estimate of $\\delta_0$ and $\\delta_1$; a testable extension is to compare the achieved residual set against the bound $D_0$ from the Lyapunov inequality across independent experiments.","Making the algorithm reproducible requires specifying the undefined vector function $\\Phi_N(z)$ and resolving the implicit appearance of the control $u$ inside the virtual control laws, which the manuscript leaves open."],"forward_implications":["If the practical prescribed-time stability proof is correct, a control engineer can fix the settling time $T$ during design, with that time independent of initial conditions and of the adaptation gain.","The same controller would apply to any strict-feedback system for which EDMDc returns a bounded-uncertainty linear model, not just the simulated Van der Pol oscillator.","The final residual set shrinks as the model-error bounds $\\delta_0$ and $\\delta_1$ shrink, so richer dictionaries and more data directly buy tighter steady-state accuracy.","Boundedness of all closed-loop signals would extend to the control input, making the scheme a candidate for safety-critical nonlinear systems despite the data-driven identification step.","By replacing repeated differentiation of virtual controls with simple derivatives in the lifted linear space, the approach lowers the computational burden of backstepping in high-dimensional lifted spaces."],"supporting_citations":[{"why":"Supplies the prescribed-time control setting and a data-driven prescribed-time convergence benchmark for the problem definition.","marker":"[2]"},{"why":"Provides the adaptive backstepping design that the proposed PTAB controller builds on and extends.","marker":"[12]"},{"why":"Shows prescribed-time adaptive backstepping applied to an uncertain system, serving as the structural template for the recursive construction.","marker":"[13]"},{"why":"States that a prescribed-time settling time can be user-defined independently of initial conditions, a property adopted by the control objective.","marker":"[17]"},{"why":"Gives the extended dynamic mode decomposition with control method used to identify the lifted linear model from data.","marker":"[19]"},{"why":"Supports the bounded-uncertainty modeling assumption by handling Koopman model error in a control setting.","marker":"[20]"},{"why":"Introduces practically prescribed-time stability, the exact notion of convergence the paper proves.","marker":"[22]"}],"fun_headline_variants":["Koopman backstepping: choose your own settling time","Data-driven control with a built-in deadline","Prescribed-time stabilization without a precise model","Adaptive backstepping in Koopman coordinates, on time","Lyapunov meets data: user-defined convergence time"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on knowing before the controller runs that all error left by the learned linear model, namely the Koopman approximation, the state reconstruction, and the external disturbance, never exceeds a known straight-line bound in the size of the lifted state; if that bound is wrong or unavailable, the convergence argument stops.","fun_headline_variants_meta":{"raw":{"variants":["Koopman backstepping: choose your own settling time","Data-driven control with a built-in deadline","Prescribed-time stabilization without a precise model","Adaptive backstepping in Koopman coordinates, on time","Lyapunov meets data: user-defined convergence time"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00041,"raw_usage":{"total_tokens":2167,"prompt_tokens":1032,"completion_tokens":1135,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":648,"completion_tokens_details":{"reasoning_tokens":1058}},"tokens_in":648,"tokens_out":1135,"duration_ms":11758,"temperature":1.0,"reasoning_tokens":1058,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:26:01.229607+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Write the closed loop down literally: because the virtual controls in (9) and (11) each contain $(Bu)_i$, the control $u$ appears on both sides of its own definition, and the function $\\Phi_N(z)$ in (12a)-(12b) is never defined; a concrete test is to attempt the substitution and see whether an explicit expression for $u$ exists at all.","supporting_citations":[{"cited_title":"Data-driven control with prescribed-time convergence for discrete-time nonlinear systems,","cited_arxiv_id":null,"evidence_quote":"Supplies the prescribed-time control setting and a data-driven prescribed-time convergence benchmark for the problem definition."},{"cited_title":"Adaptive predefined- time prescribed performance control for trajectory tracking of tailless aircraft with backstepping method,","cited_arxiv_id":null,"evidence_quote":"Provides the adaptive backstepping design that the proposed PTAB controller builds on and extends."},{"cited_title":"Prescribed-time adaptive backstepping control of an uncertain nonlinear 2-DOF helicopter,","cited_arxiv_id":null,"evidence_quote":"Shows prescribed-time adaptive backstepping applied to an uncertain system, serving as the structural template for the recursive construction."},{"cited_title":"Strong prescribed- time stabilization of uncertain nonlinear systems by periodic delayed feedback,","cited_arxiv_id":null,"evidence_quote":"States that a prescribed-time settling time can be user-defined independently of initial conditions, a property adopted by the control objective."},{"cited_title":"Towards reliable data-based optimal and predictive control using extended DMD,","cited_arxiv_id":null,"evidence_quote":"Gives the extended dynamic mode decomposition with control method used to identify the lifted linear model from data."},{"cited_title":"Model predictive control with model error compensation by Koopman approach,","cited_arxiv_id":null,"evidence_quote":"Supports the bounded-uncertainty modeling assumption by handling Koopman model error in a control setting."},{"cited_title":"Prescribed-time control with linear decay for nonlinear systems,","cited_arxiv_id":null,"evidence_quote":"Introduces practically prescribed-time stability, the exact notion of convergence the paper proves."}],"review_version":1}