{"id":"9a7545f3-1d7c-40c0-ad6d-59158ab89f8a","arxiv_id":"2506.09273","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A Gaussian-process-based hybrid internal model regulator learns the steady-state control map online and is claimed to achieve practical output regulation for a class of nonlinear systems.","lead":"The paper proposes a data-driven controller that learns the internal model of a nonlinear system online using Gaussian process regression, without needing a model-based observer. The authors claim the closed-loop system stays bounded and regulates the output to a small error that shrinks as the learned model improves.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 is not proven: Eq. (29) uses Lemma 1's high-probability GP error bound as a deterministic constant, and Eq. (33) replaces realized squared error by posterior variance, so the jump-Lyapunov decrease essential to the hybrid stability claim does not follow.","rationale":"The reader's weakest_assumption concerns the training data proxies (eta, u) versus (eta*, u*), a real concern about whether the GP learns the correct map in closed loop. My stress-test focuses on a different but equally load-bearing flaw in the proof of Theorem 1: the deterministic misuse of the probabilistic GP error bound and the substitution of posterior variance for realized squared error. The two issues are related in that both concern whether the GP approximation error term in the Lyapunov analysis is actually controlled, but they are separate logical steps. The numerical examples show plausible behaviour for the specific gains and sliding windows chosen, but they do not repair the proof. A deterministic theorem cannot be concluded from a bound that holds only with probability at least 1 - delta, unless confidence levels are propagated through the hybrid stability argument. Similarly, posterior variance is not the realized error, so the jump decrease (35) is not established. These are not merely technical gaps: without the flow inequality (30) and the jump inequality (35), Theorem 1's ultimate boundedness claim has no valid proof. The reader's REJECT verdict is therefore appropriate; the paper would need a probabilistic stability theorem and a repair of the jump analysis, as well as a treatment of the proxy training data, before it could be accepted.","tokens_in":18777,"tokens_out":8971,"duration_ms":106083,"concrete_test":"Re-derive Eqs. (29)-(35) carrying the confidence level delta of Lemma 1 explicitly. Concretely, take a scalar GP with a known function f drawn from the prior, train on n and n+1 points, and compare the realized quantity (mu_{n+1}(eta)-f(eta))^2 - (mu_n(eta)-f(eta))^2 with sigma_{n+1}^2(eta)-sigma_n^2(eta). If the realized difference is positive with nontrivial frequency despite the posterior variance decreasing, Eq. (33) is false. Then check whether the probabilistic bound in Lemma 1 permits a deterministic Lyapunov decrease for all trajectories; if not, Theorem 1 must be restated probabilistically or the proof repaired.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires a Lyapunov function that decreases along flows and does not increase at jumps. The flow analysis at Eq. (29) defines C_mu = L_mu ||zeta||^2 + sqrt(beta(rho)) sigma_DS(eta) + gamma(rho) and then treats it as a deterministic constant. But Lemma 1 supplies the bound |mu(eta*) - varpi(eta*)| <= sqrt(beta) sigma + gamma only with probability at least 1 - delta; beta depends on delta and the bound fails on a set of positive probability. No union bound over data updates and trajectories is given, so the deterministic inequality (30) is unjustified. Moreover, Eq. (33) substitutes the posterior variance for the realized squared prediction error in V(g)-V. Posterior variance is an expectation conditioned on the data, not the actual error for the realized function, so a decrease in variance does not imply V(g(chi)) - V(chi) <= 0. The state-dependent term L_mu ||zeta||^2 is also absorbed into C_mu as if it were constant. These steps are load-bearing: the jump condition (35) and the ultimate bound epsilon_mu in Theorem 1 rely on them. Consequently, the deterministic stability theorem is unsupported as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies approximate output regulation for a class of nonlinear systems with an output-feedback regulator that learns the steady-state internal-model map online using Gaussian process regression. The controller is formulated as a hybrid system: continuous-time flow with a high-gain feedback term plus a saturated GP posterior mean, and discrete-time jumps at which the GP is retrained on a sliding window of measured internal-model states and control inputs. The main result, Theorem 1, claims under Assumptions 1-5 that all closed-loop solutions are bounded and converge to a set of radius epsilon_mu, which shrinks as the GP model improves. The proof uses the Lyapunov function V=exp(T-tau)(V1+e^2+(hat u-u*)^2), a flow analysis based on a GP uniform error bound from the literature, and a jump analysis based on posterior-variance decrease. Three numerical examples are provided: a Lorenz-type system, a system with a nonlinear exosystem, and a continuous fermenter.","tokens_in":19063,"tokens_out":8915,"duration_ms":94676,"significance":"If the main theorem were correct, the paper would make a useful contribution to data-driven output regulation: it removes the polynomial steady-state generator assumption, avoids the model-based observer and the time-derivative computations required by earlier GP regulators, and proposes a hybrid implementation with a sliding window that keeps the computation manageable. The three numerical examples, especially the bioreactor, illustrate the intended application domain. The flow part of the Lyapunov argument follows standard tools, and the use of a literature GP error bound is appropriate in spirit. However, the central stability proof contains several load-bearing gaps, discussed below, so the claimed result is not established as written. The significance of the paper is therefore conditional on a substantially reworked proof.","major_comments":[{"comment":"At a jump, tau=T, so V(chi)=exp(T-tau)(V1+e^2+(mu_n(eta)-u*)^2) equals V1+e^2+(mu_n(eta)-u*)^2, not exp(T)(V1+e^2+(mu_n(eta)-u*)^2) as written in Eq. (31). After the reset tau+=0, the jump difference V(g(chi))-V(chi) contains the additional positive term (exp(T)-1)(V1+e^2). The proof ignores this term, so the jump nonincrease condition (35) does not follow even if the GP error decreases. This is a load-bearing algebraic error in the hybrid Lyapunov argument.","section":"Section IV, Eq. (31)"},{"comment":"Lemma 1 is a high-probability statement: the bound |mu(eta*)-varpi(eta*)| <= sqrt(beta) sigma + gamma holds with probability at least 1-delta, with beta depending on delta and on the data. The proof treats C_mu as a deterministic constant for all time and all data updates; no union bound over the countably many jump updates or over the continuous trajectory is supplied. Moreover, C_mu contains L_mu||zeta||^2, where zeta = bar_eta + b^{-1} N e is state-dependent, so the step from (29) to the constant epsilon_mu in (30) is not justified. Without a uniform high-probability event and a smallness condition on L_mu, the ultimate boundedness claim is unsupported.","section":"Section IV, Eqs. (29)-(30)"},{"comment":"The posterior variance sigma^2(eta) is the posterior expectation of the squared prediction error, not the realized squared error (mu(eta)-u*)^2 for the particular sample varpi. Substituting the posterior variance for the generalization error epsilon_D as defined in Eq. (17) is therefore not an equality, and a decrease in posterior variance does not imply that (mu_{n+1}(eta)-u*)^2 <= (mu_n(eta)-u*)^2 along the realized trajectory. The jump nonincrease condition (35) relies on this substitution, so the jump part of the proof is not valid as written.","section":"Section IV, Eq. (33)"},{"comment":"The training pairs are the measured internal-model state eta and the measured control u, but the proof requires that mu(eta) approximate the ideal map varpi(eta*). The measured control u includes the high-gain feedback term -kp rho(e)e and the saturation function, and during the initial cycle it is purely feedback. No assumption among Assumptions 1-5 formalizes the statement in Section III-C that eta and u 'can serve as suitable proxies' for eta* and u*. Without such a uniform approximation assumption, the learned GP map is biased, and the decomposition eta = eta* + zeta used in Eq. (28) does not supply the needed bound on mu(eta)-varpi(eta*). This is load-bearing for the regulation error estimate.","section":"Section III-C and Theorem 1"},{"comment":"The set A := {0,T} x R^{nz} x R^{n_eta} x E x Theta is not compact because the Euclidean factors R^{nz} and R^{n_eta} are unbounded. As stated, A contains every z and eta component, so the distance |phi(t,j)|_A is trivially zero in those coordinates and the claimed boundedness and convergence to A are vacuous in those components. The theorem should quantify over a compact subset of the state space or define A differently.","section":"Section IV, Theorem 1"}],"minor_comments":[{"comment":"The same symbol alpha_{1z} is used for both the lower and upper bounding functions in (25a) and (25b), and the overbar on the constant c0 appears to be lost in (24). This makes the K-infinity estimates ambiguous and should be corrected.","section":"Section IV, Eqs. (24)-(25)"},{"comment":"The saturation level M = max(mu(eta)) is not defined over a set; the boundedness of mu on the relevant compact domain should be stated explicitly before saturation is introduced.","section":"Section IV, Eq. (26)"},{"comment":"There are several typographical errors, including 'contant' for 'constant' in Assumption 1, a stray 'a' in the parameter description of Example V-A, and 'an locally pre-attractive' in Definition 3. Reference [20] appears to misspell 'Goebel' as 'Goedel'. These should be corrected in revision.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"To the editor: The proposed architecture is interesting and the simulations are useful, but the proof of Theorem 1 has multiple load-bearing errors that are not local. The jump Lyapunov computation is algebraically wrong, the variance-substitution step is conceptually invalid, the probabilistic GP bound is used as a deterministic uniform bound without a union bound, and the statement of the target set A is not compact. In my view these cannot be repaired by minor editing; a fundamentally different proof, likely stochastic or high-probability in nature, would be required. The manuscript is not suitable for publication in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper has a genuinely different architecture: a sliding-window GP learns the steady-state internal model directly from proxy data, with a hybrid update that avoids the observer and its derivative from [18]. That is worth something. The numerical examples do show error reduction, and the bioreactor case is a sensible demonstration. But the main theorem as written does not follow from the proof, and the stress-test note is right on both load-bearing points.\n\nThe flow analysis uses Lemma 1's high-probability bound as a deterministic inequality: |µ−ϖ| ≤ sqrt(β)σ + γ holds only with probability at least 1−δ, and β depends on δ. No confidence level is carried through, no union bound over jumps is given, so (29)–(30) do not justify the claimed ultimate bound. The jump analysis is worse: (33) substitutes posterior variance for the realized squared prediction error. Posterior variance is an expectation conditioned on the data, not the actual error for the realized function; a decrease in variance does not imply V(g(χ))−V(χ) ≤ 0. The state-dependent term L_µ||ζ||² is also absorbed into C_µ as if it were constant. These are not cosmetic slips; the hybrid Lyapunov decrease and the ε_µ bound rest on them. The definition of ε_µ in terms of C_µ, which itself contains the GP error bound, also makes the \"shrinks as the model improves\" statement close to a restatement of the generalization bound, though the closed-loop stability claim is not forced by fitting.\n\nThere is also a gap in the learning premise: Section III-C assumes η and u are suitable proxies for η* and u*, but this is not among Assumptions 1–5 nor in Theorem 1. If the high-gain feedback dominates, the GP learns a biased map and the decomposition in (28) breaks. This is a real concern but secondary to the proof gaps.\n\nWhat is solid: the problem is meaningful, the architecture is plausible, the flow part of the Lyapunov analysis is standard, and the simulations are honest illustrations rather than overclaiming. The authors seem to know what they are doing; the errors read like haste rather than incoherence.\n\nWho this is for: researchers working on data-driven output regulation, particularly those building on Bin, Marconi, Gentilini. A serious referee should engage. My own verdict: the theorem needs to be either made probabilistic with confidence carried through, or the deterministic argument needs a different bound. As it stands, reject with a clear path to resubmission.\n\nRecommendation: send it to peer review; it deserves referee time, and the authors could plausibly repair it.","headline":"A plausible and novel GP-hybrid regulator architecture, but the main stability theorem is not proven: the proof conflates posterior variance with actual error and uses a high-probability GP bound deterministically.","tokens_in":19590,"tokens_out":1472,"would_cite":false,"duration_ms":15067,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C10","93C40","93B52","68T05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that a Gaussian process can learn the unknown internal-model steady-state map online, yielding practical output regulation with output feedback alone.","keywords":["output regulation","Gaussian process regression","internal model","data-driven control","hybrid dynamical systems","nonlinear control","output feedback","online learning"],"falsifier":"Run the regulator on a plant whose ideal feedforward $u^\\star$ is identically zero but whose initial error is large, with $k_p$ chosen so that $-k_p\\rho(e)e$ dominates the recorded input during the first sliding window; if after the first Gaussian process update the learned map $\\mu(\\eta)$ produces predictions far from zero that are not corrected by subsequent updates, then the proxying assumption is violated and the regulation error will not converge to the predicted $\\varepsilon_\\mu$.","tokens_in":18526,"feed_emoji":"🤖","tokens_out":5599,"duration_ms":50258,"temperature":0.7,"pith_summary":"This paper proposes a data-driven output regulator for a class of nonlinear systems in which the plant and exosystem are only partially known. Instead of solving the regulator equations or building a model-based observer, it uses Gaussian process regression to learn the unknown steady-state map that links the internal model state to the ideal feedforward control. The learned map is embedded in a hybrid controller that combines a high-gain stabilizing feedback with the Gaussian process prediction, and the Gaussian process model is refreshed online on a sliding window of collected input-output data. The central claim is that, under standard assumptions, the closed-loop trajectory stays bounded and converges to a compact set whose size shrinks as the Gaussian process approximation improves, so the regulation error becomes practically small. The approach matters because it removes the need for explicit model knowledge and for computing time derivatives of the learned model.","feed_headline":"Gaussian process learns the regulator's internal map from data","feed_subtitle":"Practical output regulation from output measurements alone; the error shrinks as the learned model improves and no observer is needed.","key_machinery":"The object that carries the argument is the Gaussian-process learned internal model: the pair $(\\eta,u)$ of measured internal-model state and control input are treated as proxies for the unmeasured ideal pair $(\\eta^\\star,u^\\star)$, and a Gaussian process with squared exponential kernel is trained online to approximate the steady-state map $u^\\star = \\varpi(\\eta^\\star)$. The hybrid clock with period $T$ splits the dynamics into continuous flow (where the Gaussian process prediction is held fixed) and discrete jumps (where the dataset is updated with a sliding window), which keeps the training cost bounded. The stability proof rests on the Gaussian process posterior variance monotone decrease in (34), the uniform error bound of Lemma 1, and a high-gain feedback term $k_p\\rho(e)e$ that renders the Lyapunov derivative negative on the flow set.","core_discovery":"The paper's central claim is that practical output regulation for the nonlinear system (1)–(3) can be achieved with only measurements of the regulated output, by learning the unknown continuous map $\\varpi(\\eta^\\star)$ that produces the ideal feedforward control $u^\\star$ from the internal model state $\\eta^\\star$. The regulator (18)–(19) interconnects a linear internal model $\\dot{\\eta} = M\\eta + N u$ with a Gaussian process approximation $\\hat{u} = \\mu(\\eta)$, updated at discrete jumps of a hybrid clock, plus a high-gain term $-k_p\\rho(e)e$ that stabilizes the augmented system. Theorem 1 asserts that every solution from the compact set $\\mathcal{A}$ is bounded for all $(t,j)$ and satisfies $|\\phi(t,j)|_{\\mathcal{A}} \\le \\varepsilon_\\mu$ as $(t,j) \\to \\infty \\times \\infty$, where $\\varepsilon_\\mu$ is the Gaussian process approximation error measured through the uniform bound of Lemma 1. The proof shows the Lyapunov function $V = \\exp(T-\\tau)(V_1 + e^2 + (\\hat{u}-u^\\star)^2)$ decreases along flows when the high gain dominates, and does not increase at jumps because the Gaussian process posterior variance $\\sigma^2_{n+1}(\\eta) \\le \\sigma^2_n(\\eta)$, so the learned model improves (or at least does not worsen) with each update.","pith_inferences":["A direct extension would be to multi-input plants by using vector-valued Gaussian processes for the map $\\varpi$, but the paper's Lyapunov analysis is developed for scalar control; the vector case would require a matrix-valued posterior variance monotonicity argument.","The sliding window introduces a forgetting mechanism: if the operating regime changes, old data are discarded, which may trade asymptotic accuracy for adaptivity; the paper does not quantify how the window size $P$ affects the achievable $\\varepsilon_\\mu$.","Because the high-gain term dominates the recorded control during transients, an active-data-selection policy (e.g., collecting training samples only near steady state) could accelerate learning and reduce the bias flagged in the weakest-assumption note; this is a testable modification.","The same Gaussian-process internal-model idea could be coupled with exploration in the input to guarantee the training data cover the relevant region of the internal-model state space, which the current closed-loop data collection does not explicitly ensure."],"forward_implications":["Output regulation becomes a supervised learning problem: the controller's internal model is a nonparametric function estimated from input-output data, so no explicit solution of the regulator equations is needed.","Since the Gaussian process model is updated only at discrete jump instants, the online computational cost is bounded by the sliding window size $P$, avoiding the $O(R^3)$ training cost blow-up of a growing dataset.","The regulator works from output measurements alone, so it applies to plants with modelling errors and parameter uncertainties without any observer design.","The regulation error bound $\\varepsilon_\\mu$ is directly tied to the Gaussian process prediction error, which shrinks as informative data accumulate, so the accuracy of regulation improves automatically with continued learning.","The hybrid formulation decouples learning from control: the predictor is fixed during flow and refreshed at jumps, simplifying the numerical implementation compared to continuous-time identifiers that need the time derivative of the learned model."],"supporting_citations":[{"why":"Establishes the regulator equations and the internal model property for nonlinear output regulation, providing the steady-state map that the paper learns.","marker":"[8]"},{"why":"Justifies the generic linear internal model and the existence of the steady-state map via nonlinear Luenberger observer theory.","marker":"[5]"},{"why":"Baseline identification-based adaptive internal model approach that the paper extends to nonparametric Gaussian process learning; motivates using measured state and input as proxies.","marker":"[11]"},{"why":"Prior Gaussian-process-based data-driven output regulation that requires an extended observer and time derivatives of the Gaussian process; the paper removes that requirement.","marker":"[18]"},{"why":"Supplies the uniform error bounds for Gaussian process regression (Lemma 1) that yield the bound used in the stability proof.","marker":"[26]"},{"why":"Provides the Gaussian process posterior mean and variance formulas and the matrix inversion lemma used to show posterior variance decrease.","marker":"[24]"},{"why":"Supplies the identification of posterior variance with expected Bayesian generalization error used in the jump analysis.","marker":"[25]"}],"fun_headline_variants":["GP learns the regulator's internal map from output data alone","Data-driven nonlinear regulation: no observer, GP learns the map","Output regulation via GP: learn the internal map, shrink the error","Gaussian process regulator: learn the map, drop the observer","GP-based output regulation: internal map learned from data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The training loop assumes that the measured internal model state $\\eta$ and the applied control $u$ are close enough to the ideal signals $\\eta^\\star$ and $u^\\star$ that learning on them yields the correct map; if the high-gain feedback term dominates the recorded $u$, the Gaussian process learns a biased relationship and the decomposition in equation (28) that the stability proof relies on no longer holds.","fun_headline_variants_meta":{"raw":{"variants":["GP learns the regulator's internal map from output data alone","Data-driven nonlinear regulation: no observer, GP learns the map","Output regulation via GP: learn the internal map, shrink the error","Gaussian process regulator: learn the map, drop the observer","GP-based output regulation: internal map learned from data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00026,"raw_usage":{"total_tokens":1601,"prompt_tokens":970,"completion_tokens":631,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":586,"completion_tokens_details":{"reasoning_tokens":547}},"tokens_in":586,"tokens_out":631,"duration_ms":6017,"temperature":1.0,"reasoning_tokens":547,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:52:57.039046+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the regulator on a plant whose ideal feedforward $u^\\star$ is identically zero but whose initial error is large, with $k_p$ chosen so that $-k_p\\rho(e)e$ dominates the recorded input during the first sliding window; if after the first Gaussian process update the learned map $\\mu(\\eta)$ produces predictions far from zero that are not corrected by subsequent updates, then the proxying assumption is violated and the regulation error will not converge to the predicted $\\varepsilon_\\mu$.","supporting_citations":[{"cited_title":"Output regulation of nonlinear systems,","cited_arxiv_id":null,"evidence_quote":"Establishes the regulator equations and the internal model property for nonlinear output regulation, providing the steady-state map that the paper learns."},{"cited_title":"Output stabilization via nonlinear luenberger observers,","cited_arxiv_id":null,"evidence_quote":"Justifies the generic linear internal model and the existence of the steady-state map via nonlinear Luenberger observer theory."},{"cited_title":"Approximate nonlinear regulation via identification-based adaptive internal models,","cited_arxiv_id":null,"evidence_quote":"Baseline identification-based adaptive internal model approach that the paper extends to nonparametric Gaussian process learning; motivates using measured state and input as proxies."},{"cited_title":"Data-driven output regula- tion via Gaussian Processes and Luenberger internal models,","cited_arxiv_id":null,"evidence_quote":"Prior Gaussian-process-based data-driven output regulation that requires an extended observer and time derivatives of the Gaussian process; the paper removes that requirement."},{"cited_title":"Uniform error bounds for gaussian process regression with application to safe control,","cited_arxiv_id":null,"evidence_quote":"Supplies the uniform error bounds for Gaussian process regression (Lemma 1) that yield the bound used in the stability proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Gaussian process posterior mean and variance formulas and the matrix inversion lemma used to show posterior variance decrease."},{"cited_title":"Vivarelli,Studies on the Generalisation of Gaussian Processes and Bayesian Neural Networks","cited_arxiv_id":null,"evidence_quote":"Supplies the identification of posterior variance with expected Bayesian generalization error used in the jump analysis."}],"review_version":1}