{"id":"774163fa-ce1d-4183-bdbd-afe4ed90a997","arxiv_id":"2505.03346","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A nonlinear extension of Willems' Fundamental Lemma via Koopman bilinear realization enables direct data-driven predictive control without EDMD-based system identification.","lead":"This paper derives a data-only representation of trajectories for nonlinear control systems that can be lifted into a Koopman bilinear form, and uses it to build a predictive controller that skips model identification. The added value for a generalist: a path toward controlling unknown nonlinear systems directly from measured data, though it still needs hand-chosen lifting functions and solves a nonconvex optimization problem.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2's converse is only valid under a quadratic coupling constraint, so the DeePC problem remains nonconvex and no exact-KBR equivalence is established; this reinforces the conditional verdict based on the restrictiveness of Assumption 1.","rationale":"The paper's theoretical core, Lemma 2, is internally consistent: under full row rank of G_L, any trajectory can be lifted and represented, and the reverse direction holds when the nonlinear coupling condition is imposed. However, the coupling condition is not a linear constraint on g; it equates Vg with the Kronecker product of the data combinations Z_{1,L}g and U_{1,L}g. This means the DeePC formulation (16)-(16d) is a nonconvex bilinear program even in the ideal exact-KBR setting. The authors acknowledge the lack of global optimality guarantees in Section IV, but this admission undercuts the 'improved optimality' comparison: the reported costs are only local solutions of a nonconvex problem, and no theorem shows that solving (16) with λ_z = 0 is equivalent to solving the original MPC (9). Combined with the acknowledged rarity of exact finite-dimensional KBRs and the absence of formal error bounds for the inexact case, the central claim of a broadly applicable direct data-driven nonlinear control method is only weakly supported. The reader's conditional verdict therefore stands; my concrete check would determine whether the exact-KBR equivalence actually holds numerically.","tokens_in":11375,"tokens_out":14715,"duration_ms":150277,"concrete_test":"Settle the equivalence claim by reproducing Section IV.B on system (18) with basis [x1, x2, x1^2], a sufficiently long persistently exciting data set, L = N, and λ_z = 0. Solve (16) with a global nonconvex solver (or a dense grid for small N) and compare the returned cost and trajectory with the true MPC (9) solved to global optimality for the same horizon and initial condition. If the DeePC optimum does not match the global MPC optimum, or if the returned g fails Vg = (Z_{1,L}g) ⊗ (U_{1,L}g) to machine precision, then the exact-KBR equivalence asserted in the paper does not hold and the nonconvexity of the coupling constraint is a genuine limitation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem is internally consistent, but the converse in Lemma 2 is not a linear fundamental-lemma parameterization. The condition v[1,L] = V_{1,L,T-L+1}g in (12) is only sufficient when v_i = z_i ⊗ u_i is additionally enforced, where z_i and u_i are themselves the data combinations Z_{1,L}g and U_{1,L}g. Thus the admissible set of g is the quadratic variety Vg = (Z_{1,L}g) ⊗ (U_{1,L}g), not a subspace. The DeePC formulation (16) encodes this through the bilinear constraints (16c)-(16d), so even under Assumption 1 and exact persistency of excitation, the optimization is nonconvex. The paper gives no proof that a global solution of (16) is found, nor that (16) with λ_z = 0 is equivalent to the original nonlinear MPC (9). Consequently, the claims of 'improved optimality' rest on the local solutions returned by fmincon, and the inexact-KBR robustness claims in Section IV.C remain heuristic. This is a sharper version of the reader's concern: the barrier is not only that exact finite-dimensional KBRs are rare, but that even when one exists, the data-driven reformulation does not recover a convex or globally tractable problem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends Willems' Fundamental Lemma to nonlinear control-affine systems using the Koopman bilinear realization (KBR). Under an exact finite-dimensional KBR assumption and a persistency-of-excitation condition, it claims that any input/state trajectory of the nonlinear system can be represented by data Hankel matrices of lifted states, inputs, and bilinear products (Lemma 2). It then proposes a Data-Enabled Predictive Control (DeePC) formulation with bilinear constraints and regularization to handle inexact KBRs, and presents three numerical studies (Van der Pol oscillator with inexact KLR, a system with an exact KBR, and a system with neither exact KLR nor exact KBR) comparing against EDMD-based Koopman MPC and KLR-based DeePC. The KLR-based case is discussed as a convex special case.","tokens_in":11647,"tokens_out":8005,"duration_ms":74194,"significance":"If the main result were correct as stated, the paper would provide a meaningful step toward direct data-driven predictive control for nonlinear systems, bypassing explicit Koopman model identification. The KBR-based fundamental lemma is a nontrivial extension of the bilinear fundamental lemma of [13] and generalizes the KLR-based result in [18]. The numerical examples, while limited, show plausible performance gains. However, the exact-KBR assumption is restrictive, the converse in Lemma 2 is incomplete as stated, and the nonconvex DeePC problem lacks global optimality guarantees, so the 'improved optimality' and 'advanced robustness' claims are stronger than the evidence supports. With corrections and more careful claims, the contribution could be useful to the data-driven control community.","major_comments":[{"comment":"The converse statement in Lemma 2(ii) is not valid as stated. Condition (12) with v[1,L] = V_{1,L,T-L+1}g is a linear equation, but for (Z_{1,L+1}g, U_{1,L}g) to be a valid trajectory of the KBR (7), the additional input sequence v[1,L] must equal z_i ⊗ u_i where z_i and u_i are the corresponding entries of Z_{1,L}g and U_{1,L}g. The proof of (ii) only checks the block equation [O_L, P_L, Q_L][z(1); u; v] = Z_{2,L+1}g; it does not verify the quadratic coupling v_i = z_i ⊗ u_i for the recursively defined states. The full-row-rank condition in Definition 2 guarantees existence of a g for part (i) but does not imply that every g satisfying (12) lies in the required variety. The DeePC formulation (16) correctly includes the missing coupling as constraint (16d), so the algorithmic content is preserved, but the lemma as stated overclaims the fundamental-lemma equivalence. Please restrict the converse to g satisfying v_{1,L} = (Z_{1,L}g) ⊗ (U_{1,L}g) and adapt the proof accordingly.","section":"Section III.A, Lemma 2(ii)"},{"comment":"The DeePC problem (16) is nonconvex due to the bilinear constraints (16d), and the paper provides no global optimality guarantee. Even under the exact-KBR Assumption 1, it is not established that a global solution of (16) is equivalent to the original nonlinear MPC (9). The claim of 'improved optimality' in the abstract and in Table II is therefore an empirical observation based on local solutions returned by fmincon, not a theoretical result. The discrepancy between KB-MPC and KB-DeePC in Table II for (N,kc)=(20,20) is direct evidence of the gap. Please state precisely what (16) computes (e.g., a KKT point) and either provide an equivalence analysis under exact KBR or soften the optimality claims to 'local optimality in the lifted data space'.","section":"Section III.B, Eq. (16); Section IV.B"},{"comment":"The abstract's claim of 'advanced robustness to finite Koopman approximation errors' is supported only by two numerical examples (Sections IV.A and IV.C) without any formal analysis. The paper itself states in Section III.A that a formal treatment of the approximation error e∞_k is left to future work. While empirical demonstration is a legitimate contribution, the robustness claim should be explicitly qualified as heuristic, or the authors should provide an error bound or robustness certificate for the regularized DeePC solution. As written, the claim is stronger than the evidence provided.","section":"Section IV.C; Abstract"}],"minor_comments":[{"comment":"The dimensions of the block matrices P_L and Q_L appear to be misprinted: with B ∈ R^{nz×nu} and H ∈ R^{nz×nz nu}, one should have P_L ∈ R^{nzL×nuL} and Q_L ∈ R^{nzL×nz nu L}, not the dimensions stated in the text.","section":"Section III.A, proof of Lemma 2"},{"comment":"The objective in (16a) writes z^T Q z + u^T R u and then mentions coefficients q and r, but the linear terms q^T z and r^T u from the original MPC (9) are not present in (16a). Please either include them or remove the reference to q and r.","section":"Section III.B, Eq. (16)"},{"comment":"The notation v_{[1,T]} is used before it is defined; please explicitly define v_i = z_i ⊗ u_i for i = 1,...,T in the text preceding Definition 2.","section":"Definition 2"},{"comment":"The caption says 'The black curves represent the open-loop trajectories,' but the other curves are not fully labeled in the caption; please make the figure legend self-contained by identifying which color corresponds to which method.","section":"Section IV.A, Figure 2"}],"recommendation":"major_revision","confidential_remarks":"The paper is clearly written and honestly acknowledges the restrictiveness of Assumption 1 and the lack of formal error analysis. The main technical novelty relative to [13] is the use of the lifted state z_i in the quadratic coupling, which is a modest extension; the authors should be asked to clarify the incremental contribution relative to both [13] and [18]. The nonconvexity issue and the gap between Lemma 2(ii) and the DeePC formulation should be addressed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know about this paper. The core result, Lemma 2, is a legitimate formal extension of Yuan and Cortés's bilinear fundamental lemma to control-affine nonlinear systems that admit an exact finite-dimensional Koopman bilinear realization (KBR). The proof is a lift-and-project of the bilinear case, not a deep new idea, but it is clean, and the paper properly credits the prior work. The larger caveat is that the converse direction of Lemma 2 is not a linear parameterization: g must satisfy the quadratic coupling condition v_i = z_i ⊗ u_i, where z_i and u_i are themselves linear functions of g. The stress-test note is right: even under exact KBR and exact persistency of excitation, the DeePC problem is nonconvex because of constraints (16c)-(16d). The authors acknowledge this in Remark 2 and in Section IV.B, so it is not a hidden flaw, but it does mean the 'direct data-driven' method still relies on local solutions from fmincon and carries no global optimality or equivalence guarantee.\n\nWhat the paper does well: it identifies a real gap—EDMD-based Koopman MPC requires identification first, and existing KLR-based direct methods inherit the limitations of linear realizations—and it constructs a coherent alternative around the KBR. Remark 1 correctly recovers the KLR case as a special case, and the paper is upfront about Assumption 1 being restrictive. The three case studies are suggestive and the baselines are reasonable. The citation pattern is honest; no obvious over-claiming of novelty.\n\nThe soft spots are in the strength of the empirical claims. No code or data are provided, the 'improved optimality' claim rests on total closed-loop costs from a nonconvex solver, and the 'advanced robustness' results in Section IV.C are heuristic, with formal error analysis deferred. The paper's language in the abstract is stronger than what the evidence supports.\n\nThis is a paper for researchers working on fundamental lemma extensions or Koopman-based predictive control. It deserves a serious referee, but a referee should push for either reproducible numerical support or more tempered claims about optimality and robustness. My recommendation: send it to review, with the expectation that the authors either soften the abstract or provide the code and a more rigorous treatment of the inexact case.","headline":"A legitimate formal extension of the fundamental lemma to Koopman bilinear realizations, but the converse is a nonlinear condition, so the control problem stays nonconvex and the numerical claims are stronger than the evidence.","tokens_in":12171,"tokens_out":5675,"would_cite":true,"duration_ms":48333,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C10","93B30","93C55","37N35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a nonlinear extension of the classical Fundamental Lemma: for systems with an exact Koopman bilinear realization, persistently exciting input/state data represent every trajectory, enabling direct data-driven predictive…","keywords":["data-enabled predictive control","Koopman bilinear realization","fundamental lemma","nonlinear control-affine systems","persistency of excitation","data-driven control","Extended Dynamic Mode Decomposition"],"falsifier":"For a system without an exact KBR, such as (8) with basis $\\{x_1,x_2,\\cos x_1,\\sin x_1\\}$, collect one long persistently exciting dataset and compute, for a later trajectory $(\\bar{x},\\bar{u})$, the minimum residual $\\min_g \\|[X;U]g-[\\bar{x};\\bar{u}]\\|$ subject to $v=Vg$; a residual above numerical noise demonstrates that Lemma 2's premise of exactness is required, so the method's success then rests on regularization rather than the theorem.","tokens_in":11207,"feed_emoji":"🎛️","tokens_out":11644,"duration_ms":103854,"temperature":0.7,"pith_summary":"This paper tries to establish that predictive control for nonlinear systems can be designed directly from raw input/state data, without first identifying an approximate model. It does this by extending the classical Fundamental Lemma of behavioral systems theory to nonlinear control-affine systems, using the Koopman bilinear realization: if an exact finite-dimensional lifting exists and the collected data are persistently exciting, any possible trajectory can be written as a linear combination of data Hankel matrices. On that foundation the authors build a data-enabled predictive control formulation that works in the lifted state space and uses slack variables and regularizers to tolerate the approximation error that appears when no exact finite-dimensional Koopman realization exists. A sympathetic reader would care because this removes the EDMD identification step and, in the reported examples, achieves lower total control cost and better robustness than existing Koopman-based methods.","feed_headline":"New lemma lets data alone drive nonlinear predictive control","feed_subtitle":"Lifting to a bilinear form lets one data set represent every trajectory, skipping system identification.","key_machinery":"The central object is the Koopman bilinear realization (KBR), a lifting $z=\\Psi(x)$ such that the lifted state evolves as $z_{k+1}=A z_k+B u_k+H(z_k\\otimes u_k)$ with $x_k=C z_k$. The argument rides on the block Hankel matrix $G_L(T)=[Z,U,V]$, where $V$ is the Hankel matrix of the bilinear products $z_i\\otimes u_i$; full row rank of $G_L(T)$ is the persistency-of-excitation condition in Definition 2. Lemma 2 uses the identity $Z=[O_L,P_L,Q_L]G_L(T)$ to show that every $L$-long trajectory is a data combination, and conversely the constraint $v=Vg$ keeps combinations inside the nonlinear behavior. In the DeePC formulation, the same Hankel matrices become equality constraints that replace the unknown dynamics.","core_discovery":"The paper's central discovery is Lemma 2: under Assumption 1, if input/state data from the nonlinear system are L-persistently exciting in the sense of Definition 2, then every L-long input/state trajectory can be written as the product of the data Hankel matrices $[X;U]$ with some vector $g$, and conversely any $g$ that respects the bilinear-product constraint $v=Vg$ produces a valid trajectory. The condition requires the block Hankel matrix $[Z;U;V]$ built from lifted states, inputs, and products $z\\otimes u$ to have full row rank. When the Koopman bilinear term is zero, the constraint on $v$ disappears and the lemma specializes to the exact Koopman-linear case, recovering the classical fundamental lemma as well as earlier nonlinear extensions. The authors then formulate the DeePC problem in the lifted state, with the data representation replacing the explicit dynamics and a slack variable absorbing approximation error, and report numerical evidence that this direct formulation improves optimality over EDMD-based MPC and improves robustness over input-output Koopman DeePC when the realization is inexact.","pith_inferences":["Because Assumption 1 holds exactly when the chosen observable space is invariant under the Koopman operator, a practical route to widening the method's valid regime is to learn the lifting functions from data alongside the Hankel representation, rather than fixing them in advance.","The residual of the constraint $v=Vg$ on fresh data could serve as a measurable certificate of how far a system is from admitting an exact KBR, letting the regularization strength be tuned online.","The bilinear DeePC problem is non-convex, so global optimality is not guaranteed; alternating-minimization or branch-and-bound treatments of the constraint $v_i=z_i\\otimes u_i$ could restore the kind of certificates available in the linear case.","The rank condition in Definition 2 suggests a testable trade-off: as the lifting dimension $n_z$ and input dimension $n_u$ grow, the amount of persistently exciting data needed for full row rank may grow as well, which would determine when the direct method is practical."],"forward_implications":["For any control-affine system with an exact finite-dimensional KBR, all open-loop input/state behaviors are encoded in one persistently exciting data set, so no EDMD identification or explicit model is required for trajectory generation.","A predictive controller can be designed directly in the lifted space, where the equality constraints replace the dynamics and choosing the lifting functions to include the cost and constraints turns the nonlinear MPC problem into an optimization with bilinear constraints.","When the system admits an exact Koopman linear realization, the proposed lemma and DeePC reduce to the classical linear case, making the new result a strict generalization of both the classical Fundamental Lemma and the exact-KLR nonlinear DeePC.","In systems without an exact KBR, formulating the DeePC in the lifted input/state space is claimed to be more robust than input-output Koopman DeePC, with lower total cost in the reported Van der Pol case.","When an exact KBR exists, direct KB-DeePC matches EDMD-based KB-MPC in the numerical study, confirming that the data representation does not sacrifice prediction quality."],"supporting_citations":[{"why":"States the classical Fundamental Lemma for LTI systems, the result the paper extends.","marker":"[2]"},{"why":"Supplies the direct data-driven control formulas that motivate bypassing explicit identification.","marker":"[3]"},{"why":"Introduces the original DeePC formulation that the paper adapts to the lifted nonlinear setting.","marker":"[6]"},{"why":"Provides the Fundamental Lemma for bilinear systems that Lemma 2 builds upon.","marker":"[13]"},{"why":"Establishes Koopman-based MPC, the indirect baseline the paper compares against.","marker":"[15]"},{"why":"Extends the Fundamental Lemma to nonlinear systems with Koopman linear embedding, recovered as a special case.","marker":"[18]"},{"why":"Argues that bilinear Koopman realizations predict better than linear ones, motivating the paper's choice.","marker":"[19]"},{"why":"Constructs exact Koopman bilinear realizations for control-affine systems and supplies the example of Section IV-B.","marker":"[20]"},{"why":"Defines EDMD, the system-identification step the paper's direct method aims to avoid.","marker":"[21]"},{"why":"Reviews the regularizers used in behavioral data-driven control, justifying the penalties in (16).","marker":"[25]"}],"fun_headline_variants":["Bilinear lifting enables direct data-driven nonlinear control","Skip system ID: Koopman lemma powers nonlinear DeePC","Data alone controls nonlinear systems via bilinear Koopman","Nonlinear DeePC without EDMD: a Koopman bilinear lemma","Willems' lemma extended to nonlinear control via bilinear Koopman"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result stands on Assumption 1, that the chosen finite set of lifting functions yields an exact Koopman bilinear realization, and since the authors note most nonlinear systems do not admit such a realization, the formal guarantee covers only systems whose chosen observables form an invariant subspace.","fun_headline_variants_meta":{"raw":{"variants":["Bilinear lifting enables direct data-driven nonlinear control","Skip system ID: Koopman lemma powers nonlinear DeePC","Data alone controls nonlinear systems via bilinear Koopman","Nonlinear DeePC without EDMD: a Koopman bilinear lemma","Willems' lemma extended to nonlinear control via bilinear Koopman"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000616,"raw_usage":{"total_tokens":2838,"prompt_tokens":902,"completion_tokens":1936,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":518,"completion_tokens_details":{"reasoning_tokens":1848}},"tokens_in":518,"tokens_out":1936,"duration_ms":13185,"temperature":1.0,"reasoning_tokens":1848,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:53:29.669318+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a system without an exact KBR, such as (8) with basis $\\{x_1,x_2,\\cos x_1,\\sin x_1\\}$, collect one long persistently exciting dataset and compute, for a later trajectory $(\\bar{x},\\bar{u})$, the minimum residual $\\min_g \\|[X;U]g-[\\bar{x};\\bar{u}]\\|$ subject to $v=Vg$; a residual above numerical noise demonstrates that Lemma 2's premise of exactness is required, so the method's success then rests on regularization rather than the theorem.","supporting_citations":[{"cited_title":"A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition,","cited_arxiv_id":null,"evidence_quote":"Defines EDMD, the system-identification step the paper's direct method aims to avoid."},{"cited_title":"Formulas for data-driven control: Stabi- lization, optimality, and robustness,","cited_arxiv_id":null,"evidence_quote":"Supplies the direct data-driven control formulas that motivate bypassing explicit identification."},{"cited_title":"Data-enabled predictive con- trol: In the shallows of the DeePC,","cited_arxiv_id":null,"evidence_quote":"Introduces the original DeePC formulation that the paper adapts to the lifted nonlinear setting."},{"cited_title":"Data-driven optimal control of bilinear systems,","cited_arxiv_id":null,"evidence_quote":"Provides the Fundamental Lemma for bilinear systems that Lemma 2 builds upon."},{"cited_title":"Willems’ fundamental lemma for nonlinear systems with Koopman linear embedding,","cited_arxiv_id":null,"evidence_quote":"Extends the Fundamental Lemma to nonlinear systems with Koopman linear embedding, recovered as a special case."},{"cited_title":"Bilinearization, reachability, and opti- mal control of control-affine nonlinear systems: A koopman spectral approach,","cited_arxiv_id":null,"evidence_quote":"Constructs exact Koopman bilinear realizations for control-affine systems and supplies the example of Section IV-B."}],"review_version":1}