{"id":"0f1dcca8-6e73-47fc-91ab-09cb9af669b3","arxiv_id":"2411.10359","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The paper extends SafEDMD-based controller design to continuous-time sum-of-squares stabilization and to closed-loop performance guarantees such as a bounded L2-gain.","lead":"A data-driven control framework for unknown nonlinear systems is extended: a Koopman-based bilinear model with certified error bounds is used to design controllers with formal stability and performance guarantees. The new parts are continuous-time sum-of-squares stabilizing controllers and performance criteria such as a bounded L2-gain.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The advertised closed-loop guarantees hinge on the assumed proportional projection-error bound in Proposition 1; the numerical section sets the uncertainty constants to feasible values rather than deriving or verifying them.","rationale":"The reader's weakest_assumption correctly identifies Proposition 1's proportional projection-error bound as the load-bearing assumption. I agree: every theorem (Theorem 3, Theorem 4, Theorem 7, Corollary 9) is derived under the residual bound (21), and the SOS conditions (22)/(24)/(34) contain c_x and c_u as inputs. The paper explicitly acknowledges that deriving the projection constants is open, and the numerical section selects c_x=c_u based on LMI feasibility rather than on any data-driven or analytical verification. This is a real gap in the end-to-end certificate, not a stylistic issue: without a computable or verifiable projection bound, the controller cannot be claimed to stabilize the true nonlinear system with probability 1−δ. The gap is fixable (e.g., by computing constants for a chosen dictionary or by adding a verification step), so a CONDITIONAL verdict is appropriate rather than REJECT. The algebraic SOS derivations and the performance extension appear plausible, and the code availability is a positive; the concern is specifically that the certified guarantee is not yet connected to the actual system in the numerical study.","tokens_in":22597,"tokens_out":9410,"duration_ms":86913,"concrete_test":"For the Section 6 example, compute the exact residual r(x,u) = (σ \\hat Φ)(x) − [A \\hat Φ(x) + B0 u + \\tilde B(u⊗\\hat Φ(x))] using the known dynamics (35) and the SafEDMD estimates A, B0, \\tilde B from d=200 samples, evaluated on a dense grid over X=[−1,1]^2 and u∈{0,1}. Determine the smallest c_x,c_u such that ||r(x,u)|| ≤ c_x ||\\hat Φ(x)|| + c_u ||u|| holds pointwise on the grid. Also compute the learning-error bounds \\bar c_x,\\bar c_u from [30, Prop. 5] with d=200, δ=0.05 and compare them with the assumed values 0.1 and 0.006. If the empirically required constants exceed the assumed ones (or if the learning bound alone exceeds them), the numerical certificate is invalid as stated; if they remain below, the example's guarantee is supported, though the general method still lacks a computable projection-error bound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that a data-driven SafEDMD surrogate plus an SOS controller certifies exponential stability and quadratic performance for the unknown nonlinear system. The certificate chain depends on the residual bound (21) in Proposition 1 with constants c_x = c~_x + \\bar c_x and c_u = c~_u + \\bar c_u. The data-dependent parts \\bar c_x, \\bar c_u are covered by cited results, but the projection-error constants c~_x, c~_u are merely assumed to exist; the paper states that deriving them is 'an interesting and challenging problem for future research.' In the numerical section (Section 6), c_x and c_u are chosen as 0.1 (continuous) and 0.006 (discrete) because these are the largest values for which the comparison LMI controllers are feasible. No computation or estimate is provided showing that the actual residual r(x,u) satisfies (21) with these constants for the dictionary (36), the d=200 data points, or the sampling rate Δt=0.01. Thus the SOS feasibility in (22)/(24)/(34) certifies stability/performance only for a hypothetical uncertainty set, not for the true residual. The guarantees are therefore conditional on an unverified assumption, which is the load-bearing step between data and closed-loop certificate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a tutorial overview and extension of the SafEDMD framework for data-driven control of unknown control-affine nonlinear systems. A bilinear surrogate model of the lifted Koopman dynamics is estimated from data, and a proportional residual bound of the form ||r(x,u)|| <= c_x ||Phi_hat(x)|| + c_u ||u|| is used to synthesize feedback controllers via sum-of-squares (SOS) optimization. The authors prove exponential stabilization guarantees in continuous and discrete time (Theorems 3 and 4) and extend the approach to quadratic performance specifications such as an L2-gain bound (Theorem 7 and Corollary 9). A numerical example compares the proposed SOS-based controllers with earlier LMI-based designs and shows larger guaranteed regions of attraction and feasible uncertainty constants.","tokens_in":22927,"tokens_out":7543,"duration_ms":77121,"significance":"If the key proportional error bound can be certified from data, the framework would provide a rigorous and practical pipeline from raw data to closed-loop stability and performance certificates for unknown nonlinear systems. The paper's explicit contribution is the continuous-time SOS controller design and the performance-oriented extension, which improves on the earlier LMI-based SafEDMD designs. The paper is transparent about its assumptions and provides proofs in the appendices, and the authors share code for the numerical experiments. However, the advertised end-to-end guarantee is currently conditional on an unverified assumption on the dictionary projection error, which substantially limits the strength of the claims. The numerical experiments do not close this gap because the uncertainty constants are chosen by hand rather than computed or validated.","major_comments":[{"comment":"The load-bearing proportional projection-error bound with constants c~_x and c~_u is assumed, not derived. The paragraph immediately after the proof of Proposition 1 states that deriving rigorous proportional bounds on the projection error is 'an interesting and challenging problem for future research.' Since c_x = c~_x + \\bar c_x and c_u = c~_u + \\bar c_u enter every SOS condition (22), (24), and (34), the closed-loop stability and performance guarantees are conditional on an unverified assumption about the chosen dictionary. The paper should either provide a method to compute or certify c~_x and c~_u for a concrete class of dictionaries, or state the main theorems explicitly as conditional on a verifiable certificate and indicate how that certificate would be obtained.","section":"Section 3.3, Proposition 1 and Eq. (21)"},{"comment":"The numerical experiments do not validate the key assumption behind the guarantees. The constants c_x = c_u are chosen in (37) as the largest values for which the comparison LMI controllers from [30,31] are feasible, rather than computed from the dictionary (36), the d = 200 data points, or the sampling period \\Delta t = 0.01. No evidence is shown that the actual residual r(x,u) of the SafEDMD surrogate satisfies (21) with these constants on X = [-1,1]^2 with probability 0.95. Consequently, the guaranteed regions of attraction in Fig. 1 and the guaranteed L2-gain curve in Fig. 2 are certificates for a hypothetical uncertainty set, not for the simulated nonlinear system (35). The authors should compute the required constants from the error analysis or provide a rigorous finite-sample verification of the proportional bound (e.g., by bounding the worst-case ratio ||r(x,u)||/(||Phi_hat(x)|| + ||u||) over X and the input range).","section":"Section 6, Eq. (37) and Figs. 1-2"},{"comment":"Theorem 7 assumes that the proportional error bound (21) holds for all x in R^n, whereas Proposition 1 only provides such a bound on a compact sampling set X with probability 1 - delta. Corollary 9 is the statement whose hypotheses match the SafEDMD setting, but the paper presents Theorem 7 as the main performance result and only later qualifies it. This presentation can mislead readers into thinking that global performance guarantees are available from the data-driven framework. The paper should restructure Section 5 so that the local theorem (Corollary 9) is the main data-driven performance statement, with Theorem 7 clearly labeled as an idealized auxiliary result whose global hypothesis must be verified separately.","section":"Section 5, Theorem 7 and Corollary 9"}],"minor_comments":[{"comment":"The dictionary (36) contains the term x_2 - lambda/(lambda - 2 rho) x_1^2, which uses exact knowledge of the system parameters lambda and rho. Since the system is assumed unknown, this choice cannot be made in a purely data-driven procedure. The remark about the example's tutorial value does not remove the concern that the numerical demonstration is partly model-informed.","section":"Section 6, Remark 10 and dictionary (36)"},{"comment":"Theorem 4 is quoted from [34] without a proof in this paper. For a tutorial-style contribution this is acceptable, but the paper should state more explicitly which parts are new and which are restatements of prior work, so that the reader can distinguish the survey content from the original contributions.","section":"Section 4.2, Theorem 4"},{"comment":"The comparison between continuous-time and discrete-time SOS controllers in Fig. 1 uses different uncertainty constants (0.1 vs. 0.006), so the comparison of the regions of attraction is not apples-to-apples. This should be stated in the text or the figure caption.","section":"Section 6, Fig. 1"},{"comment":"The paper cites [39] as showing that a nonlinear data-informed dictionary admits a proportional error bound. Since this is directly relevant to the main assumption, the authors should state the concrete class of dictionaries and the resulting constants, or at least include the relevant result in a remark, so that the reader can see how the assumption can be satisfied in practice.","section":"After Proposition 1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is largely a tutorial survey of the authors' own SafEDMD framework, with the genuinely new content being the continuous-time SOS stabilization and the performance-oriented extension. The central weakness is the unverified projection-error constant: the paper explicitly defers its derivation to future work, yet uses it in every theorem and chooses it by hand in the numerics. This is a load-bearing gap that should be addressed before publication. The paper would be strengthened by either adding a section on computing or certifying c~_x and c~_u for a nontrivial dictionary, or by empirically verifying the bound on the actual residual for the reported examples. The current claims in the abstract and introduction are stronger than what is actually established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a competent, clearly written paper that extends the authors' SafEDMD line to continuous-time SOS stabilization and to performance certificates (quadratic performance, L2-gain). The math is structurally sound, the proofs are readable, and the limitation about the projection-error bound is stated plainly rather than hidden. If you work on data-driven control with certificates, it's worth a careful read.\n\nWhat's new: Theorem 3 extends the discrete-time SOS design from [34] to continuous time, reducing conservatism versus the LMI design of [30]. Section 5 (Theorem 7, Corollary 9) is genuinely new: it adds quadratic performance and L2-gain guarantees on top of SafEDMD, which the existing literature mostly lacks. The tutorial framing also does a service by assembling the SafEDMD architecture and error decomposition in one place.\n\nSoft spots. The load-bearing point is Proposition 1: the residual bound (21) depends on constants c_x = c~_x + \\bar c_x and c_u = c~_u + \\bar c_u. The learning-error parts are backed by cited results, but the projection-error constants c~_x, c~_u are assumed to exist; the paper says deriving them is future research. The numerical example then picks c_x = c_u as the largest values for which the comparison LMI controllers are feasible, with no evidence that the actual residual satisfies (21) with those constants. That means the experiments demonstrate 'if the bound holds, the SOS approach gives a larger RoA than the LMI approach,' not that the bound has been verified for the data. The reader's report captures this accurately, and I agree it's a specific, fixable weakness: compute or estimate the projection bound from the dictionary and data, or at least frame the experiment as a conditional certificate.\n\nI don't see formal circularity. The paper leans on the authors' own SafEDMD and SOS papers, but those results are published and the proofs are available; that's normal for a research program extending its own framework.\n\nWho it's for: researchers in Koopman-based control and data-driven nonlinear control with formal certificates. A referee should engage seriously: the new results are plausible and the conditional guarantees are clearly stated. On balance, this deserves peer review, with the expectation that the numerical evidence for the projection bound is tightened before acceptance.","headline":"Solid tutorial-plus-extension; the new continuous-time SOS and performance results are plausible and useful, but the advertised end-to-end guarantee is conditional on an unverified projection-error bound.","tokens_in":23401,"tokens_out":2450,"would_cite":true,"duration_ms":22567,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C10","93D15","37N35"],"pacs":[],"model":"deepseek-v4-flash","headline":"A data-driven Koopman method returns controllers with provable closed-loop stability and performance guarantees for unknown nonlinear systems.","keywords":["Koopman operator","extended dynamic mode decomposition","SafEDMD","data-driven control","sum-of-squares optimization","closed-loop guarantees","nonlinear stabilization","L2-gain"],"falsifier":"Pick a specific dictionary and system, compute the actual residual $r(x,u)$ on a dense grid, and check whether the ratio $\\|r(x,u)\\vert/(\\|\\hat\\Phi(x)\\|+\\|u\\|)$ stays bounded as $(x,u)\\to(0,0)$; if the ratio grows without bound, the proportional bound (21) fails and the SOS certificate loses its validity. Alternatively, simulate the certified controller on the true system from initial conditions inside $\\Omega(c^*)$ and look for a trajectory that leaves $\\Omega(c^*)$ or fails to converge within the $1-\\delta$ probability set, which would directly contradict Theorem 3.","tokens_in":22404,"feed_emoji":"🎛️","tokens_out":8659,"duration_ms":76279,"temperature":0.7,"pith_summary":"This paper aims to turn raw state-input data into a feedback controller for an unknown nonlinear system together with a mathematical certificate that the controller works. The approach learns a finite-dimensional bilinear surrogate model of the system's Koopman operator through the SafEDMD procedure, which supplies a guaranteed bound on the model's residual error that is proportional to the lifted state norm and the input norm. Using that bound as uncertainty, sum-of-squares optimization produces a lifted polynomial feedback law that provably renders the origin exponentially stable with a certified convergence rate, and the same construction enforces performance criteria such as a bound on the $\\mathcal L_2$-gain from disturbances to user-chosen outputs. If the claims hold, safety-critical systems can be controlled from data alone, with probabilistic but certified closed-loop guarantees rather than heuristic ones.","feed_headline":"Data alone now yields provably stabilizing controllers","feed_subtitle":"Rigorous error bounds turn Koopman learning into certified closed-loop control with guaranteed region of attraction and L2-gain.","key_machinery":"The load-bearing construction is the certified bilinear surrogate model (19), $(\\sigma\\hat\\Phi)(x)=A\\hat\\Phi(x)+B_0u+\\tilde B(u\\otimes\\hat\\Phi(x))+r(x,u)$, with the residual confined to the proportional set $\\|r(x,u)\\|\\le c_x\\|\\hat\\Phi(x)\\|+c_u\\|u\\|$ of (21). This particular shape lets a matrix-valued sum-of-squares inequality—condition (22) in continuous time, (24) in discrete time, (34) with performance—imply a genuine Lyapunov inequality for $V(x)=\\hat\\Phi(x)^\\top P^{-1}\\hat\\Phi(x)$ via the S-procedure, using the multiplier $\\tau(\\hat\\Phi(x))$. The controller $\\mu(x)=L(\\hat\\Phi(x))P^{-1}\\hat\\Phi(x)$ is the feedback that makes the closed-loop lifted dynamics $A+B_0K(z)+\\tilde B(K(z)\\otimes z)$ fit into that inequality, and $\\Omega(c^*)$ is the largest sublevel set of $V$ inside the sampling region $X$, chosen so that the data-certified error bound remains valid along all trajectories.","core_discovery":"The paper's central claim is Theorem 3: whenever the sum-of-squares condition (22) is feasible for the bilinear surrogate model $\\dot{\\hat\\Phi}(x)=A\\hat\\Phi(x)+B_0u+\\tilde B(u\\otimes\\hat\\Phi(x))+r(x,u)$ together with the certified proportional residual bound $\\|r(x,u)\\le c_x\\|\\hat\\Phi(x)\\|+c_u\\|u\\|$ of Proposition 1, the lifted polynomial feedback $\\mu(x)=L(\\hat\\Phi(x))P^{-1}\\hat\\Phi(x)$ exponentially stabilizes the origin of the unknown control-affine system (1) for every initial condition in the Lyapunov level set $\\Omega(c^*)$ contained in the sampling region $X$, with probability $1-\\delta$ over the training data. Theorem 7 and Corollary 9 extend the same machinery to the disturbed system (26), guaranteeing exponential stability together with a quadratic performance inequality, for example a finite $\\mathcal L_2$-gain from the disturbance $w$ to the performance output $y$. The discrete-time counterpart, Theorem 4, obtains the same stability guarantee with a rational sampled-data controller and needs only state-successor measurements rather than state derivatives. What makes these statements non-vacuous is the proportional shape of the residual bound: because the bound vanishes at the origin, the Lyapunov argument closes at $(x,u)=(0,0)$, so the certificate covers a neighborhood of the equilibrium rather than only a compact set.","pith_inferences":["If the open problem of deriving practical proportional projection bounds is resolved—the paper points to kernel-EDMD results—the framework becomes a fully automatic data-to-certificate pipeline with no hand-chosen error constants.","The robust invariant set $\\Omega(c^*)$ produced by the certificate could be reused as a terminal set or constraint set in model predictive control, connecting this design to the MPC extensions the authors mention.","A natural stress test is to benchmark the certified $\\mathcal L_2$-gain against the true nonlinear gain on systems with known finite-dimensional Koopman invariant subspaces; the gap would quantify the remaining conservatism of the SOS approach.","The discrete-time performance extension is left for future work; combining the rational controller parametrization of Theorem 4 with the performance matrix (34) would plausibly yield sampled-data $\\mathcal L_2$-gain certificates."],"forward_implications":["A user who collects the required state-input samples can obtain, for any confidence $1-\\delta$, a feedback law with a certified exponential convergence rate and a certified region of attraction, without ever identifying the drift or input maps $f$ and $g_i$.","The SOS-based design tolerates larger residual bounds than the earlier LMI-based SafEDMD controllers, so it remains feasible with fewer data samples and returns a larger guaranteed region of attraction (Section 6, Figure 1).","By choosing performance output matrices, the same SOS program certifies a quadratic supply-rate inequality—e.g., a guaranteed $\\mathcal L_2$-gain bound—for the closed-loop nonlinear system, provided trajectories stay inside the sampling region.","In discrete time, the sampled-data controller needs no derivative measurements, only state-successor pairs, and still guarantees exponential stability of the continuous-time system between sampling instants.","The learning-error part of the bound shrinks as $\\mathcal O(1/\\sqrt{\\delta d})$ (plus $\\Delta t^2$ in discrete time), so the guarantees can be sharpened by collecting more data with a small sampling period."],"supporting_citations":[{"why":"Introduces the Koopman operator, the infinite-dimensional linear representation that the surrogate model approximates.","marker":"[7]"},{"why":"Provides the extended dynamic mode decomposition regression used to fit the surrogate matrices from data.","marker":"[19]"},{"why":"Develops the continuous-time SafEDMD architecture, supplying the certified bilinear surrogate and residual bound used in Theorem 3.","marker":"[30]"},{"why":"Develops the discrete-time SafEDMD architecture, supplying the derivative-free surrogate model and error bound used in Theorem 4.","marker":"[31]"},{"why":"Derives the SOS-based discrete-time stabilization condition that this paper extends and whose controllers it compares against.","marker":"[34]"},{"why":"Shows that a nonlinear data-informed dictionary admits a proportional error bound, validating the assumed form of the projection bound in Proposition 1.","marker":"[39]"},{"why":"Establishes the proportional error-bound structure for data-driven MPC with practical stability, motivating the residual shape used throughout.","marker":"[28]"}],"fun_headline_variants":["Koopman learning with certified error bounds for stabilization","Provable stabilization from data via Koopman and SOS","Guaranteed closed-loop control learned purely from data","Certified Koopman controllers with stability guarantees","Data-driven Koopman control with rigorous error bounds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The main load-bearing premise is that the surrogate's residual stays below a known bound that scales linearly with the lifted state and the input, with concrete constants; the paper notes that deriving such proportional bounds for the projection error in practice is an open problem, and the numerical examples choose those constants by hand rather than computing them from data.","fun_headline_variants_meta":{"raw":{"variants":["Koopman learning with certified error bounds for stabilization","Provable stabilization from data via Koopman and SOS","Guaranteed closed-loop control learned purely from data","Certified Koopman controllers with stability guarantees","Data-driven Koopman control with rigorous error bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000449,"raw_usage":{"total_tokens":2288,"prompt_tokens":995,"completion_tokens":1293,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":611,"completion_tokens_details":{"reasoning_tokens":1217}},"tokens_in":611,"tokens_out":1293,"duration_ms":8293,"temperature":1.0,"reasoning_tokens":1217,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:42:16.103966+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick a specific dictionary and system, compute the actual residual $r(x,u)$ on a dense grid, and check whether the ratio $\\|r(x,u)\\vert/(\\|\\hat\\Phi(x)\\|+\\|u\\|)$ stays bounded as $(x,u)\\to(0,0)$; if the ratio grows without bound, the proportional bound (21) fails and the SOS certificate loses its validity. Alternatively, simulate the certified controller on the true system from initial conditions inside $\\Omega(c^*)$ and look for a trajectory that leaves $\\Omega(c^*)$ or fails to converge within the $1-\\delta$ probability set, which would directly contradict Theorem 3.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Koopman operator, the infinite-dimensional linear representation that the surrogate model approximates."},{"cited_title":"A data-driven approximation of the Koopman operator: Ex- tending dynamic mode decomposition.Journal of Nonlinear Science, 25(6):1307–1346, 08 2015","cited_arxiv_id":null,"evidence_quote":"Provides the extended dynamic mode decomposition regression used to fit the surrogate matrices from data."},{"cited_title":"Koopman-based feedback design with stability guarantees.IEEE Transactions on Auto- matic Control, 70(1):355–370, 2025","cited_arxiv_id":null,"evidence_quote":"Develops the continuous-time SafEDMD architecture, supplying the certified bilinear surrogate and residual bound used in Theorem 3."},{"cited_title":"SafEDMD: A certified learn- ing architecture tailored to data-driven control of nonlinear dynamical systems","cited_arxiv_id":null,"evidence_quote":"Develops the discrete-time SafEDMD architecture, supplying the derivative-free surrogate model and error bound used in Theorem 4."},{"cited_title":"Koopman-based control using sum-of-squares optimization: Improved stability guarantees and data efficiency.European Journal of Control, page 101286, 2025","cited_arxiv_id":null,"evidence_quote":"Derives the SOS-based discrete-time stabilization condition that this paper extends and whose controllers it compares against."},{"cited_title":"Kernel-based error bounds of bilinear Koopman surrogate models for nonlinear data- driven control","cited_arxiv_id":null,"evidence_quote":"Shows that a nonlinear data-informed dictionary admits a proportional error bound, validating the assumed form of the projection bound in Proposition 1."},{"cited_title":"Data-driven MPC with stability guarantees using extended dynamic mode decomposition.IEEE Transactions on Automatic Control, 70(1):534–541, 2025","cited_arxiv_id":null,"evidence_quote":"Establishes the proportional error-bound structure for data-driven MPC with practical stability, motivating the residual shape used throughout."}],"review_version":1}