Data-Driven Nonlinear Regulation: Gaussian Process Learning

Martin Guay; Richard D. Braatz; Shimin Wang; Telema Harry

REVIEW 5 major objections 3 minor 32 references

Data-Driven Nonlinear Regulation: Gaussian Process Learning

T0 review · 5 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2506.09273 v1 pith:EHUH3AEN submitted 2025-06-10 eess.SY cs.DMcs.SYmath.OCnlin.AO

Telema Harry , Martin Guay , Shimin Wang , Richard D. Braatz This is my paper · ORCID

classification eess.SYcs.DMcs.SYmath.OCnlin.AO MSC 93C1093C4093B5268T05

keywords outputregulationGaussianprocessregressioninternalmodeldata-drivencontrolhybriddynamicalsystemsnonlinearfeedbackonlinelearning

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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 $\varepsilon_\mu$.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The 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.

The 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.

There 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.

What 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.

Who 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.

Recommendation: send it to peer review; it deserves referee time, and the authors could plausibly repair it.

Referee Report

5 major / 3 minor

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.

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 (5)

[Section IV, Eq. (31)] 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 IV, Eqs. (29)-(30)] 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 IV, Eq. (33)] 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 III-C and Theorem 1] 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 IV, Theorem 1] 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.

minor comments (3)

[Section IV, Eqs. (24)-(25)] 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 IV, Eq. (26)] 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.
[Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the GP error bound is imported from an external lemma and the stability proof is not a reparameterization of the fitted model.

full rationale

The claimed derivation is not circular. Theorem 1's stability proof uses a Lyapunov function and imports the Gaussian-process error bound directly from Lemma 1 (an external high-probability bound from [26]), rather than fitting C_mu to the data or defining C_mu to be the regulation error. The flow inequality (29)-(30) is obtained by standard dissipativity and Young's inequality manipulations, and the ultimate ball radius epsilon_mu = alpha_1^{-1}(b_* C_mu exp(T)) is computed from that inequality rather than assumed. The jump analysis invokes the known posterior-variance-decrease property of GP conditioning; even if the substitution in Eq. (33) is mathematically questionable (posterior variance is the expected, not realized, generalization error), that is a correctness gap, not a circular reduction. Minor self-citations ([14], [23]) are motivational and do not carry the load of the theorem. Therefore the paper is self-contained with respect to circularity concerns.

Assumptions & free parameters 4 free parameters · 8 assumptions · 0 invented entities

The ledger shows the paper's main reliance on standard nonlinear regulation machinery plus a Bayesian prior on the unknown map. The proxy-data assumption is the least grounded entry; it is asserted informally and is central to the learning loop.

free parameters (4)

GP kernel hyperparameters (σ_f^2, ℓ_i) = not reported; obtained by maximum likelihood
Chosen from training data; affect the learned map and the error bound C_μ through the kernel, but no explicit values are given.
Sliding window size P = 10 (all examples)
Determines how much data the Gaussian process uses; the theorem does not specify how P affects the bound.
Hybrid update period T = 0.1 (Example 1)
Sets the discrete update schedule; appears in the residual bound as exp(T).
Control gain k_p and shaping function ρ(e) = k_p = 500/700 (Example 1), 300 (Example 2), 30 (Example 3); ρ(e) = e^2 + 1
The theorem requires k_p sufficiently large; values are hand-tuned and appear in the flow derivative.

assumptions (8)

domain assumption Assumption 1: b(w,σ) is continuous, positive, and lower bounded by b⋆
Required for the input-output structure and for the gain condition in the Lyapunov proof.
domain assumption Assumption 2: regulator equations are solvable; z⋆(w,σ) exists
Defines the ideal steady-state and u⋆; standard in nonlinear output regulation.
ad hoc to paper Assumption 3: ϖ(η⋆) is a sample from a zero-mean Gaussian process with kernel K
Imposes a Bayesian prior on the unknown map; needed for the Gaussian process error bound Lemma 1 but hard to verify in practice.
domain assumption Assumption 4: ϖ has bounded norm in the RKHS of K
Standard in Gaussian process error bounds; needed for Lemma 1.
standard math Assumption 5: kernel K is smooth and Lipschitz
Standard; satisfied by the squared-exponential kernel used.
domain assumption The internal model (7) with controllable (M,N), M Hurwitz, n_η = 2(n_w+n_z+1) exists
Taken from [5] and [9]; underpins the augmented system.
ad hoc to paper Measured (η,u) are suitable proxies for unmeasured (η⋆,u⋆)
Introduced informally in Section III-C; load-bearing for the training data but never formalized as an assumption.
domain assumption The z-subsystem admits an ISS Lyapunov function with the supply-rate property
Used in the proof of Lemma 2; invoked via [29] and [32].

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Cite this review

Pith. "Pith review of Data-Driven Nonlinear Regulation: Gaussian Process Learning." pith.science (2026). https://pith.science/paper/EHUH3AEN

@misc{pith2026250609273,
  author       = {Pith},
  title        = {Pith review of: Data-Driven Nonlinear Regulation: Gaussian Process Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EHUH3AEN}},
  note         = {Machine review of arXiv:2506.09273}
}

read the original abstract

This article addresses the output regulation problem for a class of nonlinear systems using a data-driven approach. An output feedback controller is proposed that integrates a traditional control component with a data-driven learning algorithm based on Gaussian Process (GP) regression to learn the nonlinear internal model. Specifically, a data-driven technique is employed to directly approximate the unknown internal model steady-state map from observed input-output data online. Our method does not rely on model-based observers utilized in previous studies, making it robust and suitable for systems with modelling errors and model uncertainties. Finally, we demonstrate through numerical examples and detailed stability analysis that, under suitable conditions, the closed-loop system remains bounded and converges to a compact set, with the size of this set decreasing as the accuracy of the data-driven model improves over time.

Figures

Figures reproduced from arXiv: 2506.09273 by the authors.

**Figure 1.** This GP-based internal model learning regulator is formulated within a hybrid system framework as    τ˙ = 1, η˙ = Mη + Nu, ˆ ˙uˆ = 0, C := [0, T] × R nη × Θ (18a)    τ + = 0, η + = η, uˆ + = ϖˆ (DS ), D := {T} × R nη × Θ (18b) where the training dataset obtained by the sliding window of size P is given by DS + = (DS ∪ {(η, uˆ)}) \ {(η(tk−P ), u(tk−P ))}. The GP model is updated at each discrete time event … view at source ↗

**Figure 1.** Hybrid Framework for a Closed-Loop Control System with a Gaussian Process-Based Nonlinear Regulator: The system operates in a continuous [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 3.** Example V-A: Tracking error trajectories with and without internal [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 2.** Figure 2: Example V-A: The trajectories of the ideal steady-state control [PITH_FULL_IMAGE:figures/full_fig_p009_2.png]

**Figure 4.** Figure 4: Example V-B: Tracking error trajectories with and without internal [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Continuous bioreactor. C. Example 3: Control of a continuous fermenter Consider the anaerobic continuous bioreactor in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 7.** Figure 7: Example V-C: The upper plot illustrates the trajectory of the tracking [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 6.** Figure 6: Example V-C: Dynamic responses of the continuous bioreactor [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 8.** Figure 8: Example V-C: Closed-loop dynamic responses of the continuous [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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