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arxiv: 2511.21248 · v2 · submitted 2025-11-26 · 📡 eess.SY · cs.SY· math.OC

Stability of data-driven Koopman MPC with terminal conditions

Irene Schimperna , Lea Bold , Johannes K\"ohler , Karl Worthmann , Lalo Magni This is my paper

Pith reviewed 2026-05-17 05:14 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords data-driven MPCKoopman operatorstability analysisrecursive feasibilityterminal conditionsproportional error boundkEDMDnonlinear systems
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The pith

If a data-driven Koopman surrogate satisfies a proportional error bound, then MPC with terminal conditions asymptotically stabilizes the true nonlinear plant.

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

The paper establishes that model predictive control using a data-driven approximation of the system dynamics can still guarantee asymptotic stability of the original nonlinear system, provided the approximation error satisfies a proportional bound. This bound requires the error to grow no faster than linearly with the size of the current state and applied input. Such a condition can be met for many nonlinear systems when the surrogate model is constructed via kernel extended dynamic mode decomposition based on the Koopman operator. A sympathetic reader would care because this opens the door to using learned models in safety-critical control without requiring an exact first-principles description of the plant.

Core claim

We prove recursive feasibility and asymptotic stability of the closed loop when a proportional error bound holds for the data-driven prediction model in an MPC scheme that includes terminal costs and constraints. For a broad class of nonlinear systems this proportional bound is satisfied by Koopman models identified through kernel extended dynamic mode decomposition from data.

What carries the argument

The proportional error bound on the difference between true and predicted dynamics, which is linear in the norms of state and input, combined with terminal conditions that keep the finite-horizon optimization feasible over time.

If this is right

  • The MPC optimization problem stays feasible at every time step.
  • The plant state converges asymptotically to the origin despite using an approximate model.
  • The result applies directly to Koopman models identified from data via kEDMD for suitable nonlinear systems.
  • A numerical case study confirms that the framework can be implemented on concrete plants.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could reduce reliance on precise physical modeling when designing controllers for complex nonlinear plants.
  • Similar proportional bounds might be derived for other data-driven surrogate methods such as neural-network predictors.
  • The terminal-condition technique could be combined with constraint tightening to handle additional disturbances.

Load-bearing premise

The prediction error of the data-driven model must remain bounded by a constant times the combined norm of the state and input at each step.

What would settle it

A concrete nonlinear system where the kEDMD error is verified to be proportional yet closed-loop trajectories under the proposed MPC diverge or fail to reach the origin.

Figures

Figures reproduced from arXiv: 2511.21248 by Irene Schimperna, Johannes K\"ohler, Karl Worthmann, Lalo Magni, Lea Bold.

Figure 1
Figure 1. Figure 1: Van der Pol (22): Error ∥x(k)∥ of the kEDMD-based MPC with different numbers of clusters. We consider virtual observation points arranged in a grid of Padua points, as described in [36], to which we add a point x1 = 0. In particular, we consider d ∈ {352, 1327} virtual observation points. For each virtual observation point, we sample di = 25 data points (xij , uij , x+ ij ) ∈ BrX (xi), i ∈ [1 : d], conside… view at source ↗
read the original abstract

This paper derives conditions under which Model Predictive Control (MPC) with terminal conditions, using a data-driven surrogate model as a prediction model, asymptotically stabilizes the plant despite approximation errors. In particular, we prove recursive feasibility and asymptotic stability if a proportional error bound holds, where proportional means that the bound is linear in the norm of the state and the input. For a broad class of nonlinear systems, this condition can be satisfied using data-driven surrogate models generated by kernel Extended Dynamic Mode Decomposition (kEDMD) using the Koopman operator. Last, the applicability of the proposed framework is demonstrated in a numerical case study.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript derives conditions for asymptotic stability of Model Predictive Control (MPC) with terminal conditions when the prediction model is a data-driven Koopman surrogate obtained via kernel Extended Dynamic Mode Decomposition (kEDMD). It proves recursive feasibility and asymptotic stability whenever the approximation error satisfies a proportional bound ||e(x,u)|| ≤ γ(||x|| + ||u||) with γ sufficiently small relative to the terminal set and cost. The paper asserts that this bound holds for a broad class of nonlinear systems when the surrogate is generated by kEDMD, and demonstrates the framework on a numerical case study.

Significance. If the proportional error bound is rigorously established for kEDMD surrogates, the result would supply a practical route to stability certificates for data-driven Koopman MPC that avoids the need for exact linearization or perfect models. The manuscript supplies explicit proofs of recursive feasibility and asymptotic stability under the stated error condition, together with a terminal-cost and terminal-set construction that is standard yet carefully adapted to the error setting. These elements constitute a clear technical contribution provided the kEDMD applicability claim is substantiated.

major comments (2)
  1. [§4] §4 (or the section asserting kEDMD applicability): the claim that finite-data kEDMD residuals satisfy ||e(x,u)|| ≤ γ(||x|| + ||u||) for a broad class of nonlinear systems is stated without an explicit derivation relating the operator approximation error, kernel choice, and sampling density to the linear scaling constant γ. The residual of a finite Gram-matrix projection generally contains a bias term that need not vanish linearly at the origin; a concrete bound or sufficient condition on data density and kernel parameters is required to make the second half of the central claim load-bearing rather than an assertion.
  2. [Theorem 1 and Theorem 2] Theorem 1 (recursive feasibility) and Theorem 2 (asymptotic stability): both results condition on the proportional error bound holding with γ small enough relative to the terminal set radius and the decrease rate of the terminal cost. Because the paper does not derive γ from the kEDMD construction, the stability guarantee remains conditional on an external hypothesis whose satisfaction is not verified within the manuscript.
minor comments (2)
  1. Notation for the error term e(x,u) should be introduced once and used consistently; currently the proportional bound is written with varying symbols across the stability section and the kEDMD discussion.
  2. The numerical case study would benefit from an explicit plot or table reporting the observed ||e(x,u)|| / (||x|| + ||u||) ratio over the state-input domain to illustrate that the proportional bound is attained in practice.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the constructive referee report. We have carefully considered the major comments and provide point-by-point responses below. We plan to make revisions to address the concerns regarding the substantiation of the kEDMD error bound.

read point-by-point responses

  1. Referee: [§4] §4 (or the section asserting kEDMD applicability): the claim that finite-data kEDMD residuals satisfy ||e(x,u)|| ≤ γ(||x|| + ||u||) for a broad class of nonlinear systems is stated without an explicit derivation relating the operator approximation error, kernel choice, and sampling density to the linear scaling constant γ. The residual of a finite Gram-matrix projection generally contains a bias term that need not vanish linearly at the origin; a concrete bound or sufficient condition on data density and kernel parameters is required to make the second half of the central claim load-bearing rather than an assertion.

    Authors: We thank the referee for this observation. The manuscript states that the proportional error bound can be satisfied for a broad class of nonlinear systems via kEDMD, but does not provide a detailed derivation of the constant γ in terms of kernel and sampling parameters. We acknowledge that this leaves the applicability claim somewhat assertive. In the revised manuscript, we will expand §4 to include a derivation of sufficient conditions under which the kEDMD residual satisfies the proportional bound. This will involve assumptions on the reproducing kernel Hilbert space, the density of sampling points, and the smoothness of the underlying dynamics, showing that the bias term can be bounded linearly near the origin for systems where the Koopman operator has suitable spectral properties. We will also add a remark on the practical choice of kernel parameters to ensure γ is sufficiently small. revision: yes

  2. Referee: [Theorem 1 and Theorem 2] Theorem 1 (recursive feasibility) and Theorem 2 (asymptotic stability): both results condition on the proportional error bound holding with γ small enough relative to the terminal set radius and the decrease rate of the terminal cost. Because the paper does not derive γ from the kEDMD construction, the stability guarantee remains conditional on an external hypothesis whose satisfaction is not verified within the manuscript.

    Authors: We agree that Theorems 1 and 2 are conditional on the proportional error bound holding with γ small enough. This structure is intentional, as the primary contribution is the derivation of stability conditions for data-driven MPC under this error model. The link to kEDMD is made by asserting that the bound is achievable, which we will substantiate with the added analysis in §4 as described in our response to the first comment. In the revision, we will also update the abstract, introduction, and conclusion to emphasize that the stability guarantees are subject to the error bound being satisfied by the data-driven model, and that this is possible under the conditions we now derive for kEDMD. This will make the overall claim self-contained within the manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: stability result is conditional on external assumption

full rationale

The paper's core derivation proves recursive feasibility and asymptotic stability of the closed-loop MPC system whenever the data-driven model satisfies an external proportional error bound ||e(x,u)|| ≤ γ(||x|| + ||u||) with γ sufficiently small. This bound is introduced as a hypothesis rather than constructed from the MPC cost or terminal set, so the stability theorem does not reduce to a tautology. The subsequent statement that kEDMD Koopman models can realize such a bound for a broad class of nonlinear systems is an applicability claim supported by the properties of kernel EDMD; it does not feed back into the stability proof or rename a fitted residual as a derived prediction. No self-citation chain, ansatz smuggling, or uniqueness theorem imported from prior author work appears load-bearing for the main result. The derivation chain is therefore self-contained against the stated assumption.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests primarily on the domain assumption of a proportional error bound for the data-driven model and standard terminal conditions from MPC theory; no new entities are postulated and free parameters appear limited to the error-bound scaling constant.

free parameters (1)
  • proportional error bound scaling constant
    The linear error bound is assumed to hold with some scaling factor that must be satisfied or estimated for the kEDMD model.
axioms (2)
  • domain assumption Standard MPC terminal cost and terminal set satisfy the usual decrease and invariance conditions for stability
    Invoked implicitly to extend classical MPC stability results to the data-driven case.
  • domain assumption The data-driven Koopman surrogate approximates the true nonlinear dynamics with an error linear in state and input norms
    This is the load-bearing assumption stated in the abstract for the stability proof.

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Forward citations

Cited by 1 Pith paper

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  1. Discounted MPC and infinite-horizon optimal control under plant-model mismatch: Stability and suboptimality

    math.OC 2026-04 unverdicted novelty 5.0

    Exponential stability and suboptimality guarantees for discounted and undiscounted MPC under plant-model mismatch proportional to states and inputs, with uniform robustness over horizon length.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Learning-based model predictive control: Toward safe learning in control,

    L. Hewing, K. P. Wabersich, M. Menner, and M. N. Zeilinger, “Learning-based model predictive control: Toward safe learning in control,”Annual Review of Control, Robotics, and Autonomous Sys- tems, vol. 3, no. 1, pp. 269–296, 2020

    work page 2020

  2. [2]

    Stabilizing predictive control of nonlinear ARX models,

    G. De Nicolao, L. Magni, and R. Scattolini, “Stabilizing predictive control of nonlinear ARX models,”Automatica, vol. 33, no. 9, pp. 1691–1697, 1997

    work page 1997

  3. [3]

    Cautious model predictive control using gaussian process regression,

    L. Hewing, J. Kabzan, and M. N. Zeilinger, “Cautious model predictive control using gaussian process regression,”IEEE Transactions on Control Systems Technology, vol. 28, no. 6, pp. 2736–2743, 2019

    work page 2019

  4. [4]

    Learning- based model predictive control for safe exploration,

    T. Koller, F. Berkenkamp, M. Turchetta, and A. Krause, “Learning- based model predictive control for safe exploration,” in57th IEEE Conference on Decision and Control (CDC), pp. 6059–6066, 2018

    work page 2018

  5. [5]

    A tutorial review of neural network modeling approaches for model predictive control,

    Y . M. Ren, M. S. Alhajeri, J. Luo, S. Chen, F. Abdullah, Z. Wu, and P. D. Christofides, “A tutorial review of neural network modeling approaches for model predictive control,”Computers & Chemical Engineering, vol. 165, 107956, 2022

    work page 2022

  6. [6]

    Recurrent neural network-based MPC for systems with input and incremental input constraints,

    I. Schimperna, G. Galuppini, and L. Magni, “Recurrent neural network-based MPC for systems with input and incremental input constraints,”IEEE Control Systems Letters, vol. 8, pp. 814–819, 2024

    work page 2024

  7. [7]

    A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition,

    M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, “A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition,”Journal of Nonlinear Science, vol. 25, pp. 1307–1346, 2015

    work page 2015

  8. [8]

    An overview of Koopman-based control: From error bounds to closed-loop guarantees,

    R. Str ¨asser, K. Worthmann, I. Mezi ´c, J. Berberich, M. Schaller, and F. Allg ¨ower, “An overview of Koopman-based control: From error bounds to closed-loop guarantees,”Annual Reviews in Control, vol. 61, 101035, 2026

    work page 2026

  9. [9]

    Spectral properties of dynamical systems, model reduction and decompositions,

    I. Mezi ´c, “Spectral properties of dynamical systems, model reduction and decompositions,”Nonlinear Dynamics, vol. 41, pp. 309–325, 2005

    work page 2005

  10. [10]

    Spectral analysis of nonlinear flows,

    C. W. Rowley, I. Mezi ´c, S. Bagheri, P. Schlatter, and D. S. Henningson, “Spectral analysis of nonlinear flows,”Journal of Fluid Mechanics, vol. 641, pp. 115–127, 2009

    work page 2009

  11. [11]

    Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control,

    M. Korda and I. Mezi ´c, “Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control,”Auto- matica, vol. 93, pp. 149–160, 2018

    work page 2018

  12. [12]

    Koopman form of nonlinear systems with inputs,

    L. C. Iacob, R. T ´oth, and M. Schoukens, “Koopman form of nonlinear systems with inputs,”Automatica, vol. 162, 111525, 2024

    work page 2024

  13. [13]

    Finite- data error bounds for Koopman-based prediction and control,

    F. N ¨uske, S. Peitz, F. Philipp, M. Schaller, and K. Worthmann, “Finite- data error bounds for Koopman-based prediction and control,”Journal of Nonlinear Science, vol. 33, 14, 2023

    work page 2023

  14. [14]

    Towards reliable data-based optimal and predictive control using extended DMD,

    M. Schaller, K. Worthmann, F. Philipp, S. Peitz, and F. N ¨uske, “Towards reliable data-based optimal and predictive control using extended DMD,”IFAC-PapersOnLine, vol. 56, no. 1, pp. 169–174, 2023

    work page 2023

  15. [15]

    Kernel-based approximation of the Koopman generator and Schr ¨odinger operator,

    S. Klus, F. N ¨uske, and B. Hamzi, “Kernel-based approximation of the Koopman generator and Schr ¨odinger operator,”Entropy, vol. 22, 722, no. 7, 2020

    work page 2020

  16. [16]

    L∞-error bounds for approximations of the Koopman operator by kernel extended dynamic mode decomposition,

    F. K ¨ohne, F. M. Philipp, M. Schaller, A. Schiela, and K. Worthmann, “L∞-error bounds for approximations of the Koopman operator by kernel extended dynamic mode decomposition,”SIAM Journal on Applied Dynamical Systems, vol. 24, no. 1, pp. 501–529, 2025

    work page 2025

  17. [17]

    Kernel- based Koopman approximants for control: Flexible sampling, error analysis, and stability,

    L. Bold, F. M. Philipp, M. Schaller, and K. Worthmann, “Kernel- based Koopman approximants for control: Flexible sampling, error analysis, and stability,”SIAM Journal on Control and Optimization,

    work page

  18. [18]

    Accepted for publication, preprint arxiv:2412.02811

    work page arXiv

  19. [19]

    J. B. Rawlings, D. Q. Mayne, M. Diehl,et al.,Model Predictive Control: Theory, Computation, and Design. Nob Hill Publishing, 2017

    work page 2017

  20. [20]

    Kouvaritakis and M

    B. Kouvaritakis and M. Cannon,Model predictive control. Springer, 2016

    work page 2016

  21. [21]

    Input-to-state stable MPC for constrained discrete-time nonlinear systems with bounded additive uncertainties,

    D. Lim ´on Marruedo, T. Alamo, and E. F. Camacho, “Input-to-state stable MPC for constrained discrete-time nonlinear systems with bounded additive uncertainties,” in41st IEEE Conference on Decision and Control (CDC), vol. 4, pp. 4619–4624, 2002

    work page 2002

  22. [22]

    A novel constraint tightening approach for nonlinear robust model predictive control,

    J. K ¨ohler, M. A. M ¨uller, and F. Allg ¨ower, “A novel constraint tightening approach for nonlinear robust model predictive control,” inIEEE Annual American Control Conference (ACC), pp. 728–734, 2018

    work page 2018

  23. [23]

    Nominally robust model predictive control with state constraints,

    G. Grimm, M. J. Messina, S. E. Tuna, and A. R. Teel, “Nominally robust model predictive control with state constraints,”IEEE Transac- tions on Automatic Control, vol. 52, no. 10, pp. 1856–1870, 2007

    work page 2007

  24. [24]

    Offset-free MPC ex- plained: novelties, subtleties, and applications,

    G. Pannocchia, M. Gabiccini, and A. Artoni, “Offset-free MPC ex- plained: novelties, subtleties, and applications,”IFAC-PapersOnLine, vol. 48, no. 23, pp. 342–351, 2015

    work page 2015

  25. [25]

    Offset-free nonlinear MPC with Koopman-based surrogate models,

    I. Schimperna, L. Bold, and K. Worthmann, “Offset-free nonlinear MPC with Koopman-based surrogate models,”IFAC-PapersOnLine,

    work page

  26. [26]

    Accepted for publication, preprint arxiv:2504.10954

    work page arXiv

  27. [27]

    Data-driven model predictive control: Asymptotic stability despite approximation errors exemplified in the Koopman framework,

    I. Schimperna, K. Worthmann, M. Schaller, L. Bold, and L. Magni, “Data-driven model predictive control: Asymptotic stability despite approximation errors exemplified in the Koopman framework,”arXiv preprint arXiv:2505.05951, 2025

    work page arXiv 2025

  28. [28]

    Beyond inherent robustness: strong stability of MPC despite plant-model mismatch,

    S. J. Kuntz and J. B. Rawlings, “Beyond inherent robustness: strong stability of MPC despite plant-model mismatch,”IEEE Transactions on Automatic Control, pp. 1–13, 2025

    work page 2025

  29. [29]

    Data-driven MPC with terminal conditions in the Koopman frame- work,

    K. Worthmann, R. Str ¨asser, M. Schaller, J. Berberich, and F. Allg¨ower, “Data-driven MPC with terminal conditions in the Koopman frame- work,” in63rd IEEE Conference on Decision and Control (CDC), pp. 146–151, 2024

    work page 2024

  30. [30]

    Data-driven MPC with stability guarantees using extended dynamic mode decom- position,

    L. Bold, L. Gr ¨une, M. Schaller, and K. Worthmann, “Data-driven MPC with stability guarantees using extended dynamic mode decom- position,”IEEE Transactions on Automatic Control, vol. 70, no. 1, pp. 534–541, 2025

    work page 2025

  31. [31]

    Kernel EDMD for data-driven nonlinear Koopman MPC with stability guar- antees,

    L. Bold, M. Schaller, I. Schimperna, and K. Worthmann, “Kernel EDMD for data-driven nonlinear Koopman MPC with stability guar- antees,”IFAC PapersOnLine, 2025. Accepted for publication, preprint arxiv:2501.08709

    work page arXiv 2025

  32. [32]

    Gr ¨une and J

    L. Gr ¨une and J. Pannek,Nonlinear model predictive control. Springer, 2017

    work page 2017

  33. [33]

    Extensions of the path-integral formula for computation of Koopman eigenfunctions,

    S. A. Deka and U. Vaidya, “Extensions of the path-integral formula for computation of Koopman eigenfunctions,” in63rd IEEE Conference on Decision and Control (CDC), pp. 2875–2881, 2024

    work page 2024

  34. [34]

    When Koopman meets Hamilton and Jacobi,

    U. Vaidya, “When Koopman meets Hamilton and Jacobi,”IEEE Transactions on Automatic Control, pp. 1–16, 2025

    work page 2025

  35. [35]

    Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators,

    G. Fasshauer and Q. Ye, “Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators,” Numerische Mathematik, vol. 119, pp. 585–611, 2011

    work page 2011

  36. [36]

    Approximate interpolation with appli- cations to selecting smoothing parameters,

    H. Wendland and C. Rieger, “Approximate interpolation with appli- cations to selecting smoothing parameters,”Numerische Mathematik, vol. 101, pp. 729–748, 2005

    work page 2005

  37. [37]

    On excitation of control-affine systems and its use for data-driven Koopman approximants,

    P. Schmitz, L. Bold, F. M. Philipp, M. Rosenfelder, P. Eberhard, H. Ebel, and K. Worthmann, “On excitation of control-affine systems and its use for data-driven Koopman approximants,”arXiv preprint arXiv:2511.03734, 2025

    work page internal anchor Pith review arXiv 2025

  38. [38]

    Bivariate polynomial interpolation on the square at new nodal sets,

    M. Caliari, S. De Marchi, and M. Vianello, “Bivariate polynomial interpolation on the square at new nodal sets,”Applied Mathematics and Computation, vol. 165, no. 2, pp. 261–274, 2005

    work page 2005