Kernelized offset-free data-driven predictive control for nonlinear systems

Mircea Lazar; Thomas Oliver de Jong

REVIEW 4 major objections 4 minor 19 references

Kernelized offset-free data-driven predictive control for nonlinear systems

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that a kernelized velocity-form data-driven predictive controller can achieve offset-free tracking for unknown nonlinear systems, with recursive feasibility and Lyapunov stability guarantees under an exactness assumption.

desk verdict Useful data-driven MPC extension with a clean least-squares learning step, but the stability proof has a load-bearing gap that needs repair before the guarantees can be trusted. read the letter →

arxiv 2411.18762 v1 pith:NGMSPNN6 submitted 2024-11-27 eess.SY cs.SY

Thomas Oliver de Jong , Mircea Lazar This is my paper · ORCID

classification eess.SYcs.SY

keywords data-drivenpredictivecontroloffset-freekernelmethodsvelocityformmodelnonlinearsystemsrecursivefeasibilitystabilitysequentialquadraticprogramming

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

The paper tries to show that a data-driven predictive controller for nonlinear systems can be made offset-free without knowing the plant model, by learning a velocity (incremental) representation with kernel functions. Preserving the structure of an analytic velocity model reduces the learning step to a linear least squares problem, which is far simpler than the nonlinear optimization used in comparable Koopman-based approaches. The resulting controller, called vKDPC, is a nonlinear program that can be solved with sequential quadratic programming, and the paper gives terminal conditions under which recursive feasibility and Lyapunov stability hold. On a pendulum benchmark, the data-driven controller tracks piecewise-constant references with zero offset after disturbances, matching an analytic velocity-form NMPC in performance. If true, this provides a practical route to offset-free predictive control for unknown nonlinear systems with stability guarantees.

What carries the argument

The central object is the velocity form (incremental) state-space model, where the state is $z_k = \mathrm{col}(y_{k-1}, \Delta x_k)$ and the dynamics matrices $A$, $B$, $C$ depend nonlinearly on scheduling variables. The paper's key move is to parameterize each unknown gradient matrix ($\nabla_x f$, $\nabla_u f$, $\nabla_x h$) as a product of a constant coefficient matrix and a kernel basis matrix built from the data. Because these matrices enter linearly, the representer theorem applies and the coefficients are obtained by least squares. This kernelized velocity model has the same dimension as the analytic model, and the prediction matrices $\Psi$ and $\Gamma$ are formed by products of these kernel matrices, so the MPC problem can be solved with sequential QP by fixing the scheduling variables and iterating.

What would settle it

For the pendulum example, recompute the terminal set $Z_T$ from the kernelized model and test whether $A_{cl}(\rho) Z_T \subseteq Z_T$, $K Z_T \subseteq \Delta U$, and the LMI of Assumption 4.2 hold at a dense grid of points inside $Z_T$; any violation would invalidate the recursive feasibility proof, since the paper does not verify Assumption 4.2 over the whole set.

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Extended reading notes

Core claim

The central claim is that the structure of the analytic velocity form can be preserved exactly when functions are replaced by kernel-based representations, because the unknown gradient matrices appear linearly. This turns model learning into two independent least-squares problems: one for the output-gradient coefficients and one for the state/input-gradient coefficients. The learned model has the same state dimension as the analytic velocity model, so the online predictive control problem remains a parameterized nonlinear program solvable by sequential QP. The paper then states Assumption 4.2, a terminal cost and terminal set condition involving a locally stabilizing feedback and an LMI, and proves that under exact kernel representation this yields recursive feasibility and asymptotic Lyapunov stability of the closed loop. Simulations on a discretized pendulum show offset-free tracking for piecewise-constant references under piecewise-constant disturbances, with performance essentially identical to the analytic velocity-form NMPC.

Load-bearing premise

The kernelized velocity model exactly represents the true velocity dynamics; if the finite-sample kernel model is only approximate, the paper's recursive feasibility and asymptotic stability guarantees do not apply.

Editorial extensions

If this is right

Any nonlinear system whose velocity-form gradient functions lie in the RKHS of the chosen kernel can be controlled offset-free from noiseless data alone, without solving a nonlinear identification problem.
The terminal cost and terminal set conditions (Assumption 4.2) apply to both analytic and data-driven velocity-form MPC, replacing the more conservative terminal equality constraint used in earlier velocity-form NMPC.
Because the kernelized model has the same dimension as the analytic model, the online computation cost of vKDPC is comparable to that of model-based velocity-form NMPC; in the pendulum example it averaged 0.0683 s per sample versus 0.0438 s.
The learning step reduces to solving two least-squares problems, and the paper reports 2.0 s to fit a 2000-sample dataset for the pendulum, so the approach scales linearly with data size in the coefficient matrices.
When the kernel model is inexact, the guarantees degrade from asymptotic stability to input-to-state stability, and recursive feasibility can be recovered by leaving the initial extended state as a free optimization variable, per the robust framework referenced in the paper.

Reading between the lines

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

A natural next test is whether the least-squares kernel model remains accurate when the data are noisy; the paper assumes noiseless data, and the offset-free property in that setting is not established.
The same structural preservation could be applied to other basis functions, e.g., polynomials or neural features, where the linear-in-parameters property holds, potentially giving offset-free data-driven MPC beyond kernels.
The proof of recursive feasibility depends on the kernel model being exact; a practical user would need to verify Assumption 4.2's LMI over the entire terminal set, or rely on the robust variant, before deploying vKDPC safely.
Because the scheduling variables $\rho$ are fixed during each QP step, the convergence of Algorithm 1 to a local optimum of Problem 4.1 is not guaranteed; a reader could test whether the 4\text{--}5 average iterations reach the same cost as the analytic vNMPC on harder nonlinearities.

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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

Here's my read. The paper does something genuinely useful: it turns the kernelized velocity-form model from [13] into a practical offset-free DPC scheme by adding an explicit least-squares learning step and terminal ingredients for recursive feasibility and stability. The learning problem in Lemma 3.3 is clean, and the dimension-preserving structure means the online cost stays at the level of the analytic velocity model. The pendulum comparison against the analytic vNMPC is the right kind of sanity check, and the paper is honest about the exact-model assumption in the theoretical part.

The soft spot is real, though. Theorem 4.3's proof, as printed, is not a valid proof for the stated problem. The prediction model uses increments Δu, but the shifted candidate sequence is written in absolute inputs and plugged into the convolution formula as if it were built for u. The terminal step also fails to subtract the reference r before invoking invariance of Z_T, which is defined in error coordinates. Since Theorem 4.4 leans on Theorem 4.3, the formal stability guarantee isn't established. I agree with the stress-test note on this.

This is repairable: shift the increments, write the candidate tail with K(z_{N|k}-r), and show z̃_{N|k+1}-r ∈ Z_T. The proof structure is standard and I'd expect it to go through. But as it stands, the main theorem is not proven.

Two smaller issues. The 'exact approximation' assumption in Section IV is doing a lot of work, and the offset-free property is demonstrated in simulation but not proven for the stated disturbance model. The paper also cites [14] as very recent related work but doesn't benchmark against it; given how close that work is, a quantitative comparison would materially strengthen the paper. No code or data is shipped, which makes reproduction harder.

Overall: the core idea is good, the learning is sound, and the gap is fixable. The paper deserves a serious referee, but the referee should insist on a corrected proof and a robustness discussion. I wouldn't cite it in its current form.

Referee Report

4 major / 4 minor

Summary. The paper proposes a data-driven predictive control scheme for nonlinear systems based on a kernelized velocity-form state-space model. The unknown gradients in the velocity model are parameterized with kernel functions and learned by least squares, and the resulting controller is formulated as a nonlinear program solved by sequential quadratic programming. The authors state terminal cost and set conditions intended to guarantee recursive feasibility and Lyapunov stability, and they illustrate the method on a discretized pendulum example, comparing it with an analytic velocity-form NMPC.

Significance. The main idea, preserving the velocity-model structure so that a fixed-dimensional kernelized predictor can be learned by a closed-form least-squares problem, is attractive, and the simulation results are encouraging. The least-squares derivation in Lemma 3.3 is clean for noiseless data, and the authors are explicit about the exactness assumption on the learned model. If the theoretical gaps in Section IV were repaired, the paper would be a useful contribution to data-driven nonlinear MPC. As it stands, however, the recursive feasibility and stability claims are not established as written, and the numerical example does not verify the assumptions under which those claims are proven.

major comments (4)

[Section IV, Theorem 4.3 proof] The proof of Theorem 4.3 defines the shifted sequence tilde u_{k+1} in absolute control inputs, but the vKDPC problem in (15) optimizes over incremental inputs and the prediction matrices (13)-(14) act on Delta u. The displayed derivation then applies the convolution formula to absolute inputs, which is inconsistent with the model. The correct candidate is the incremental sequence {Delta u*_{1|k}, ..., Delta u*_{N-1|k}, K(z*_{N|k}-r)}. Consequently, recursive feasibility of Problem 4.1 at time k+1 is not established by the proof as written.
[Section IV, Assumption 4.2 and Theorem 4.3] Assumption 4.2 and the terminal step of the proof of Theorem 4.3 are not stated in consistent coordinates. The terminal constraint is (z_{N|k}-r) in Z_T, so Z_T is a set of deviations, while the proof concludes that tilde z_{N|k+1} = A_cl(rho) z*_{N|k} lies in Z_T. Since A(rho)r = r for the velocity model (2), one must instead show that A_cl(rho)(z*_{N|k}-r) lies in Z_T. The proof omits the subtraction of r and therefore does not establish the terminal constraint. Relatedly, the inclusion Z_T subset Z in Assumption 4.2 is not dimensionally consistent if Z_T is a set of deviations and Z is a set of absolute states; the condition should be {r+e : e in Z_T} subset Z, or Z should be redefined in error coordinates.
[Section IV, exactness assumption] Before Assumption 4.2, the paper assumes that the kernelized velocity dynamics hat A, hat B, hat C are an exact approximation of the true velocity dynamics. For a finite sample of noiseless data from an unknown system, this exactness assumption is generally false. Theorems 4.3 and 4.4 are proved only for this exact representation, and Remark 4.5 merely defers the inexact case to robust methods in [18]. The simulation in Section V does not verify exactness or the terminal LMI over the computed terminal set, so the theoretical guarantees do not apply to the controller actually implemented in the example. The paper should either provide a quantified approximation bound and adapt the stability analysis, or clearly present the numerical results as purely empirical.
[Section V and Remark 4.6] In Remark 4.6 and Section V, the terminal ingredients P, K, and Z_T are computed from the linearized dynamics at the reference, i.e., (hat A(x_r,u_r)+hat B(x_r,u_r)K). Assumption 4.2, however, requires the nonlinear, state-dependent LMI and set invariance to hold for all e in Z_T. No argument is given that the computed terminal set satisfies the nonlinear conditions, for example via Lipschitz bounds or a grid verification. This gap is load-bearing because the recursive feasibility and stability theorems rely on Assumption 4.2.

minor comments (4)

[Lemma 3.3] The full rank hypothesis is ill-posed: the matrices [Ky_1 ... Ky_{s-1}] and [Kx_1 ... Kx_{s-1}] have dimensions (n s) x (s-1) and ((n+m)s) x (s-1), so they cannot have full row rank for s>1; if full column rank is meant, the least-squares solution is not unique and the pseudoinverse formulas give only the minimum-norm minimizer. Please clarify.
[Theorem 4.4 proof] In the proof of Theorem 4.4, the expression (z*_{N|k}-r)(Q+K^T R K)(z*_{N|k}-r) should be a quadratic form with transposes: (z*_{N|k}-r)^T (Q+K^T R K) (z*_{N|k}-r).
[Remark 3.2] The claim of equal computational complexity should be qualified: the optimization variables have the same dimensions as in the analytic velocity model, but evaluating the kernel matrices has cost O(s) per entry.
[General presentation] There are several typos, e.g., 'dyanmics' in Remark 4.6 and 'a offset-free' in the abstract; also, Fig. 2 would benefit from labeled axes and a quantitative validation metric for the kernel model.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the kernelized model is learned from data and benchmarked against an independent analytic vNMPC controller; stability claims are conditional on standard terminal assumptions.

full rationale

The derivation chain is self-contained rather than circular. The kernelized velocity model is obtained by applying the representer theorem and solving least-squares problems (Lemma 3.3) on noiseless data, and the multi-step predictor is validated against separate test data (Fig. 1). The central closed-loop claim is an external comparison with an analytic velocity-form NMPC controller on the pendulum benchmark, so the reported offset-free performance is not forced by the fitted coefficients. The recursive-feasibility and stability results are conditional on Assumption 4.2, which postulates standard terminal cost, terminal set, and local stabilizing gain; these are assumptions used in a standard MPC argument, not conclusions derived from the data fit or from the target result itself. The paper does cite prior and same-author work, notably [2] for the velocity-model structure and [5]/[18] for incremental-input offset-free DPC and robust extensions, but those citations are not load-bearing for the claimed kernelized formulation, the least-squares learning problem, or the vNMPC comparison. A correctness gap appears to exist in the printed proof of Theorem 4.3: the shifted sequence is written in absolute inputs while the velocity model uses increments, and the terminal-invariance step does not explicitly carry the reference subtraction in the error coordinate. However, that is a proof-rigor issue rather than a circularity issue; the result does not reduce by construction to its own inputs. No self-definitional reduction, fitted-parameter-renamed-as-prediction, or author-imported uniqueness argument is present. Thus no significant circularity is found.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a small set of hand-tuned and fitted quantities, including kernel width, least-squares coefficients, and terminal ingredients, and on strong domain assumptions: noiseless data, full-rank regression, exact velocity representation, and existence of terminal ingredients. These are standard in the MPC and kernel literature but reduce the strength of the stated guarantees.

free parameters (4)

Kernel width sigma = sqrt(200)
Chosen by hand for the inverse multiquadric kernel in Section V; the learned model and all closed-loop results depend on it.
Kernel coefficient matrices A_alpha, B_alpha, C_alpha = Not reported numerically
Learned via least squares from 2000 training samples; they define the prediction matrices used in Problem 4.1.
Terminal cost P and gain K = Not reported
Computed per reference via an LMI from the kernelized model in Remark 4.6; they enter Assumption 4.2 and the stability proof.
Cost weights Q and R and horizon N = Q=1000 I3, R=10, N=20
Controller tuning choices in the simulation; they affect feasibility and terminal sets.

assumptions (5)

standard math The unknown nonlinear functions in the velocity model lie in the finite-dimensional RKHS spanned by kernel sections at the training points.
Invoked via the representer theorem before equation (5).
domain assumption Training data are noiseless and the kernel matrices have full rank.
Lemma 3.3 and Problem 2.4 state noiseless data; full rank is needed for the least-squares solutions.
domain assumption The velocity-form representation in [2, Chapter 4] is exact, with the practical approximation that the convex-hull points can be replaced by current measurements.
Exactness follows from the mean value theorem only for points in the convex hull; the model uses current measurements.
ad hoc to paper Assumption 4.2: a single terminal gain K, matrix P, and set Z_T satisfy the nonlinear LMI and set inclusions.
This is an assumed existence condition, not proved for general nonlinear systems; the example computes it only near a reference linearization.
domain assumption The disturbance in (1) is piecewise constant and does not enter the stability analysis.
Offset-free behavior is claimed for persistent disturbances, but Theorems 4.3 and 4.4 analyze only the disturbance-free velocity model.

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Pith. "Pith review of Kernelized offset-free data-driven predictive control for nonlinear systems." pith.science (2026). https://pith.science/paper/NGMSPNN6

@misc{pith2026241118762,
  author       = {Pith},
  title        = {Pith review of: Kernelized offset-free data-driven predictive control for nonlinear systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NGMSPNN6}},
  note         = {Machine review of arXiv:2411.18762}
}

read the original abstract

This paper presents a kernelized offset-free data-driven predictive control scheme for nonlinear systems. Traditional model-based and data-driven predictive controllers often struggle with inaccurate predictors or persistent disturbances, especially in the case of nonlinear dynamics, leading to tracking offsets and stability issues. To overcome these limitations, we employ kernel methods to parameterize the nonlinear terms of a velocity model, preserving its structure and efficiently learning unknown parameters through a least squares approach. This results in a offset-free data-driven predictive control scheme formulated as a nonlinear program, but solvable via sequential quadratic programming. We provide a framework for analyzing recursive feasibility and stability of the developed method and we demonstrate its effectiveness through simulations on a nonlinear benchmark example.

Figures

Figures reproduced from arXiv: 2411.18762 by the authors.

**Figure 1.** N = 20 step y prediction of kernel model for test data plotted together with the true system trajectory and the error e := y − yˆ [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Terminal set ZT for r = col(0.5, 0, 0) obtained using the kernelized velocity model (blue) and analytic velocity model (red). r changes. This amounts to 3 terminal costs for the example shown in Fig.3. For every P a corresponding control gain K is computed as explained in Remark 4.6 and a terminal set ZT is computed that fulfills the conditions from Assumption 4.2. The terminal set is computed using the MPT3 toolbox… view at source ↗

**Figure 3.** vKDPC (—) vs. vNMPC (—) trajectories and disturbance signal. VI. CONCLUSIONS In this paper we developed a novel kernelized offset-free data-driven predictive control scheme for nonlinear systems. By exploiting the structure of an analytic velocity state-space model, we reduced learning the kernelized velocity model to solving a least squares problem. The resulting offset-free kernelized DPC scheme can be efficiently… view at source ↗

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