Kernel EDMD for data-driven nonlinear Koopman MPC with stability guarantees

Irene Schimperna; Karl Worthmann; Lea Bold; Manuel Schaller

REVIEW 3 major objections 4 minor 2 cited by

Kernel EDMD for data-driven nonlinear Koopman MPC with stability guarantees

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Kernel EDMD-based model predictive control can stabilize nonlinear systems from data without terminal conditions, with practical asymptotic stability guarantees.

desk verdict Clear idea and promising error framework, but Proposition 1's perturbed cost-controllability bound is not bounded as claimed, so the main stability theorem is unproven. read the letter →

arxiv 2501.08709 v2 pith:VE73QVEX submitted 2025-01-15 math.OC

Lea Bold , Manuel Schaller , Irene Schimperna , Karl Worthmann This is my paper · ORCID

classification math.OC MSC 93D1593C1093C55

keywords kernelextendeddynamicmodedecompositionKoopmanoperatormodelpredictivecontrolpracticalasymptoticstabilitycostcontrollabilityWendlandkernelsdata-drivenerrorbounds

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's aim is a stability certificate for model predictive control (MPC) when the prediction model is a data-driven kernel EDMD surrogate of a nonlinear control-affine system. It claims that, if the true system is cost controllable and the surrogate error is pointwise proportional to the distance to the data grid, then the MPC closed loop is practically asymptotically stable, meaning it converges to a neighborhood of the origin whose size shrinks with the data density, without any stabilizing terminal cost or terminal constraint. This matters because terminal ingredients are often the hard part of nonlinear MPC design, and EDMD-based surrogates have previously needed restrictive invariance assumptions to get such guarantees. The paper proves the claim through Theorem 3 and supports it with simulations on a van der Pol oscillator.

What carries the argument

The load-bearing object is the two-level kernel EDMD surrogate: first, local linear regression at virtual cluster points $x_i$ approximates the system matrices $[g_0(x_i)\mid G(x_i)]$ from sample triples; second, Wendland-kernel interpolation lifts these approximations into a control-affine surrogate $x^+=g^\varepsilon_0(x)+G^\varepsilon(x)u$. Its key property is the pointwise error bound $\|F(x,u)-F^\varepsilon(x,u)\|_\infty\le C_1 \varepsilon_{h_X}\,\mathrm{dist}(x,X)+C_2 c\|K_X^{-1}\|\varepsilon_c$, which with $\varepsilon_c=0$ becomes proportional to $\|x\|$ because $0\in X$. Proposition 1 converts that proportional error into perturbed cost controllability, and Theorem 3 feeds the perturbed constants into the relaxed-Lyapunov framework of the nonlinear-MPC PAS framework used in [9] to obtain practical asymptotic stability.

What would settle it

For the van der Pol example with $\varepsilon_c=0$, refine the Chebyshev grid (increase $d$) and record the asymptotic offset of the closed-loop trajectory: the theorem predicts the offset goes to zero as $h_X\to 0$. If the offset does not shrink toward zero with increasing $d$, or if, for a fixed small $h_X$, the value-function decrease $\alpha$ computed from the surrogate violates $\alpha\in(0,1)$, the claimed guarantee fails.

Watch

Extended reading notes

Core claim

The central discovery is that kernel EDMD surrogates inherit cost controllability with controllable accuracy: under exact data at cluster points ($\varepsilon_c=0$) and small fill distance $h_X$, the surrogate's cost-controllability constants $B^\varepsilon_N$ converge to the nominal constants $B_N$. With a horizon $N$ chosen so that the contraction factor $\alpha_N\in(0,1)$ from the nominal cost-controllability analysis remains valid, the EDMD-based MPC controller renders the origin $\varepsilon$-practically asymptotically stable on a sublevel set of its value function, meaning every trajectory eventually enters and stays in a ball whose radius tends to zero as the fill distance $h_X$ tends to zero.

Load-bearing premise

The guarantee rests on the surrogate inheriting cost controllability with constants close enough to the nominal ones that the horizon-$N$ contraction factor $\alpha_N^\varepsilon$ in Theorem 3 stays in $(0,1)$; the paper does not quantify how small $h_X$ must be to ensure this, and the proof requires exact data at the cluster points ($\varepsilon_c=0$).

Editorial extensions

If this is right

Every control-affine system satisfying cost controllability admits a kernel EDMD surrogate for which MPC without terminal ingredients is practically asymptotically stable; the same horizon that works for the nominal system continues to work once the fill distance is small enough.
The ultimate bound on the closed-loop trajectory can be made arbitrarily small by refining the cluster-point grid, since the error bound and hence $\varepsilon$ shrink with $h_X$.
State constraints in the surrogate optimization must be tightened by the error bound, so the feasible region shrinks as data density decreases.
Because the error bound is pointwise and vanishes at the cluster points, the stability neighborhood is centered at the origin and does not require invariance of the dictionary.

Reading between the lines

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

Extension: the same proof pattern would work for any surrogate whose error is proportional to the state and uniformly continuous in the state, so the result is a template for stability certificates beyond kernel EDMD.
Extension: treating sampling noise would require a quantitative smallness condition on $\varepsilon_c$; a probabilistic version of the bound could yield high-probability practical stability with noisy data.
Extension: the exponential decay seen in the simulations suggests practical exponential stability may hold under a strengthened uniformity condition on the cost-controllability constants, which the paper does not claim.

Share X Bluesky LinkedIn Reddit HN

Formalized claims in Lean

Claim #1: The central discovery is that kernel EDMD surrogates inherit cost controllability with controllable accuracy: under exact data at cluster points ($\varepsilon_c=0$) and small fill distance $h_X$, the surrogate's cost-controllability constants $B^\varepsilon_N$ converge to the nominal constants $B_N$. With a horizon $N$ chosen so that the contraction factor $\alpha_N\in(0,1)$ from the nominal cost-

/-- @claim 1 The central discovery is that kernel EDMD surrogates inherit cost controllability with controllable accuracy: under exact data at cluster points ($\varepsilon_c=0$) and small fill distance $h_X$, the surrogate's cost-controllability constants $B^\varepsilon_N$ converge to the nominal constants $B_N$. With a horizon $N$ chosen so that the contraction factor $\alpha_N\in(0,1)$ from the nominal cost- -/
def central_claim : Prop :=

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

First thing to know: the main stability theorem is not proven in the text, and Proposition 1, which is supposed to do the heavy lifting, has a real gap. The stress-test note is right: the coefficients c_1^ε and c_2^ε grow like d^N with d = 2(L_F + Cε_hX)^2. For d > 1, which holds for the van der Pol example and for any system with L_F > 1/√2, the sequence B_k^ε is not bounded as claimed. The proof of Proposition 1 only shows a finite-horizon bound that grows exponentially in N; it does not give the uniform bound that cost controllability requires and that Theorem 3's α formula assumes. On top of that, Theorem 3 states the α condition using the nominal B_k, not the perturbed B_k^ε, and defers the proof to [3]. So as written, the claimed ε-PAS is conditional on an unquantified controllability margin.

That is a pity, because the paper has a lot going for it. The combination of kernel EDMD error bounds from [4] with the stability framework from [3] is a natural and worthwhile step, and the pointwise proportional error bound is the right tool. The idea of removing the dictionary invariance assumption is genuinely new and important for the Koopman-MPC subfield. The numerical section is honest and illustrates the expected behavior, including the offset for εc > 0.

The other limitations are less severe: the theory requires εc = 0, so exact data at cluster points; sampling noise is not covered. The proof of Lemma 1 depends on derivative error bounds that are plausible but not fully detailed. And the paper leans on several self-authored preprints, but those are independent results with their own proofs, so that is not a circularity problem.

Net: the conceptual contribution is solid and the errors are in the right place, but the central perturbation estimate is not correct as stated. The paper needs a substantial revision of Proposition 1 and Theorem 3—either by deriving a finite-horizon controllability margin that is compatible with the growth rates, or by adding a condition on L_F and C. Until then, the stability guarantee is not established.

Who should read this: anyone working on data-driven MPC with Koopman surrogates will benefit from the problem setup and the error framework, but they should treat the main theorem as open. I would send it out for review because the question matters and the authors have a credible path to fixing it, but I would not accept it as is.

Referee Report

3 major / 4 minor

Summary. The paper proposes kEDMD-MPC, a model predictive control scheme that uses a kernel-EDMD surrogate of a control-affine nonlinear system. Building on pointwise error bounds proportional to the state distance, the authors aim to prove practical asymptotic stability (PAS) of the origin for the closed loop without terminal ingredients, using cost controllability of the original system. The scheme is stated in Algorithm 1, theoretical results are given in Section 4 (Lemma 1, Proposition 1, Theorem 3), and numerical experiments on a van der Pol oscillator are reported in Section 5.

Significance. If the proof were complete, the paper would provide a useful advance: it would remove the restrictive invariance assumptions often needed in EDMD-based MPC stability proofs and would give deterministic, data-dependent stability guarantees for kernel-EDMD surrogates. The central idea of leveraging pointwise proportional error bounds and cost controllability is promising, and the numerical examples support the qualitative behavior. However, the main stability theorem depends on a cost-controllability preservation result whose proof is incomplete, and the theorem itself uses nominal rather than perturbed constants, so the claimed guarantees are not established as stated.

major comments (3)

[Section 4, Proposition 1] The boundedness of the sequence (B^ε_k) is not proven and is in fact false under the stated assumptions. In the proof, the coefficients c^ε_1 and c^ε_2 contain sums of powers of d = 2(L_F + C ε_hX)^2. When d > 1, these coefficients grow like d^N, so the expression for B^ε_N grows exponentially in N. The proposition asserts the existence of a monotonically increasing and bounded sequence (B^ε_k) but imposes no condition implying d < 1, and for the van der Pol example in Section 5 the Lipschitz constant L_F is greater than 1, so d ≈ 2L_F^2 > 1. Thus the proof does not establish the claimed bounded sequence, and the cost-controllability preservation result is not available in the form needed by Theorem 3.
[Section 4, Theorem 3] The horizon condition in Theorem 3 uses the nominal constants B_k in the formula for α_N, whereas the perturbed constants B^ε_k from Proposition 1 are what enter the stability analysis of the surrogate-based MPC loop. The argument needs to show that α^ε_N ∈ (0,1) for the chosen horizon N, using the perturbed constants, and to state a quantitative smallness condition on h_X that guarantees this. The sentence 'The proof resembles the proof of Theorem 10 in [3]' is not a substitute for this step, since the perturbed constants change the relevant inequalities. Because Proposition 1 only claims pointwise convergence B^ε_k → B_k as h_X → 0 for each fixed k, the theorem requires an additional uniformity or margin argument that is not provided.
[Section 4, Lemma 1 / Proposition 1 / Theorem 3] All the theoretical stability guarantees are proven only for εc = 0, i.e., for exact data at the virtual cluster points. The numerical section uses εc = √2/d > 0 and states that PAS 'is also obtained' in that case, but the theory does not cover this regime. This is a significant scope limitation: the advertised data-driven stability guarantee does not apply to the sampling noise or data corruption represented by nonzero cluster radius. The manuscript should either extend the theorem to εc > 0 or explicitly frame the εc > 0 behavior as a conjecture supported by simulations.

minor comments (4)

[Section 2, Theorem 2] The symbol ε is used both for the total error bound in (8) and as a perturbation parameter in definitions such as the surrogate F^ε and the admissibility set U^ε_N. This overloading makes statements like 'for all ε ∈ (0, ε0]' ambiguous, since ε in (8) is determined by h_X and εc. Please introduce a separate symbol for the error level (for example, δ or η) and use it consistently.
[Section 3, Definition 2] The quantifier order in Definition 2 is confusing: 'for each fill distance h_X and cluster radius ε_c such that (8) holds with ε ∈ (0, ε0]' suggests that h_X and ε_c are chosen after ε0, but the error bound ε depends on these quantities. Please rephrase to clarify that the existence of ε0 quantifies over sufficiently small h_X and εc.
[Section 4, Proposition 1 proof] The sentence 'Note that the first summands in J^ε_N (ˆx, u) and J_N (ˆx, u) coincide' should say 'the first summand (k=0) coincides', since the two sums are over the same k-range and the initial state is the same. This is a minor wording issue.
[Section 5, paragraph after Eq. (18)] The Lipschitz constant L_F of the van der Pol map (18) is not reported. Since the validity of the bound in Proposition 1 depends on whether d = 2(L_F + C ε_hX)^2 is less than or greater than 1, reporting L_F (or the relevant bound) would help the reader assess the example against the assumptions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stability claim is derived from proven error bounds and standard MPC robustness arguments, with only rigor gaps and self-citation burden.

full rationale

The derivation chain is not circular in the sense of reducing to its own inputs. Theorem 2's pointwise error bound is restated with a proof sketch in the paper and rests on interpolation estimates from the textbook [20] plus regression analysis from [4]; the dependence on [4] is a citation to a proof-carrying prior result, not to a fitted quantity. Proposition 1 and Lemma 1 are proven in the text, and the transfer to practical asymptotic stability via [9, Theorem 11.10] and cost controllability is a standard template whose assumptions are verified, modulo the unquantified horizon condition. The main self-citations ([3], [4], [12]) are load-bearing but not circular in the forbidden sense: they supply theorems with their own stated assumptions and proofs, and the paper does not define its surrogate or error bound in terms of the PAS conclusion. The noted weaknesses are rigor gaps rather than circularity: Theorem 3's proof is delegated to a statement that it 'resembles the proof of Theorem 10 in [3]', and Proposition 1's displayed bound has coefficients that grow like (2(L_F + C epsilon_hX)^2)^N when the Lipschitz factor exceeds one, so the asserted boundedness of (B^epsilon_k) is not actually demonstrated. Neither issue makes the claimed result equivalent to its inputs by construction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central proof rests on cost controllability of the nominal system, error bounds imported from self-authored prior work ([4], [12]), and two standard theorems from approximation theory and MPC. No new physical entities are introduced.

free parameters (5)

fill distance h_X = h_X ≤ h0 (simulations: Chebyshev grids, d=441, 1681)
The error bound (8) and the proportional bound (13) scale with h_X; the theory requires h_X below a threshold h0.
cluster radius εc = 0 in theory; sqrt(2)/d in simulations
Proposition 1 and Theorem 3 require εc=0 for the proportional error bound; positive εc introduces a constant offset in (8).
prediction horizon N = N=10 and N=20 in simulations
Theorem 3 requires N large enough that α_N ∈ (0,1); the proof of existence is not shown in the paper.
cost weights Q, R = Q=I2, R=1e-4 in simulations
Stage cost (11) enters cost controllability and the relaxed Lyapunov inequality.
control samples per cluster d_i = d_i=25 in simulations
Assumption 1 requires d_i ≥ m+1 (here m=1) and rank condition; the error bound in Theorem 2 depends on max_i ||U_i^†||.

assumptions (6)

domain assumption The true system F is control-affine, locally Lipschitz, and cost controllable on Xε with constants B_k
Proposition 1 assumes (14) for every x̂ ∈ Xε; this is the main stabilizability condition.
domain assumption State set X is convex and compact with 0 in its interior; control set U is compact with 0 in its interior
Used in Definition 1, the tightened constraint (10), and the MPC stability theorem.
domain assumption Data satisfies Assumption 1: rank([u_i1 ... u_idi]) = m and x_ij ∈ B_εc(x_i)
Defines the data requirements for the surrogate construction.
domain assumption Wendland kernel of smoothness k≥1 and F ∈ C^{ceil((n+1)/2+k)}_b(Ω; R^n)
Needed for the fill-distance error bounds in Theorem 1 and Theorem 2 and for Lemma 1.
domain assumption Ω is forward invariant w.r.t. F
Assumed in Section 2 to streamline the presentation; the paper cites [12] for the general case.
standard math Scattered data interpolation error bound [20, Theorem 11.17] and nominal MPC stability theorem [9, Theorem 11.10]
Used as black boxes in Lemma 1, Theorem 2, and Theorem 3.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Kernel EDMD for data-driven nonlinear Koopman MPC with stability guarantees." pith.science (2026). https://pith.science/paper/VE73QVEX

@misc{pith2026250108709,
  author       = {Pith},
  title        = {Pith review of: Kernel EDMD for data-driven nonlinear Koopman MPC with stability guarantees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VE73QVEX}},
  note         = {Machine review of arXiv:2501.08709}
}

read the original abstract

Extended dynamic mode decomposition (EDMD) is a popular data-driven method to predict the action of the Koopman operator, i.e., the evolution of an observable function along the flow of a dynamical system. In this paper, we leverage a recently-introduced kernel EDMD method for control systems for data-driven model predictive control. Building upon pointwise error bounds proportional in the state, we rigorously show practical asymptotic stability of the origin w.r.t. the MPC closed loop without stabilizing terminal conditions. The key novelty is that we avoid restrictive invariance conditions. Last, we verify our findings by numerical simulations.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A Data-Driven Prescribed-Time Control Framework via Koopman Operator and Adaptive Backstepping
eess.SY 2025-07 reject novelty 4.0 of 10

A Koopman-based adaptive backstepping controller is claimed to achieve prescribed-time stabilization for nonlinear systems, but the proof is incomplete and the control law contains an undefined term.
Two-component controller design to safeguard data-driven predictive control
math.OC 2025-05 conditional novelty 4.0 of 10

A two-controller architecture that uses a funnel controller to guarantee output constraints while a DeePC or EDMD-based predictive controller learns, permitting safe online data collection and tracking.

Reference graph

Works this paper leans on

25 extracted references · 20 canonical work pages · cited by 2 Pith papers

[3]

Data-driven MPC with stability guarantees using extended dynamic mode decomposition

Lea Bold, Lars Gr ¨une, Manuel Schaller, and Karl Worthmann. Data-driven MPC with stability guarantees using extended dynamic mode decomposition. IEEE Transactions on Automatic Control, 70(1):534–541, 2025

work page 2025
[4]

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

Lea Bold, Friedrich M Philipp, Manuel Schaller, and Karl Worthmann. Kernel-based Koopman approximants for control: Flexible sampling, error analysis, and stability. Preprint arXiv:2412.02811, 2024

arXiv 2024
[1]

Non- parametric control-Koopman operator learning: Flexible and scalable models for prediction and control

Petar Bevanda, Bas Driessen, Lucian Cristian Iacob, Roland Toth, Stefan Sosnowski, and Sandra Hirche. Non- parametric control-Koopman operator learning: Flexible and scalable models for prediction and control. Preprint arXiv:2405.07312, 2024

arXiv 2024
[2]

Stability and feasibility of state constrained MPC without stabilizing terminal constraints

Andrea Boccia, Lars Gr ¨une, and Karl Worthmann. Stability and feasibility of state constrained MPC without stabilizing terminal constraints. Systems & Control Letters, 72:14–21, 2014

work page 2014
[5]

Fasshauer and Q

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

work page 2011
[6]

Behavioral theory for stochastic systems? A data-driven journey from Willems to Wiener and back again

Timm Faulwasser, Ruchuan Ou, Guanru Pan, Philipp Schmitz, and Karl Worthmann. Behavioral theory for stochastic systems? A data-driven journey from Willems to Wiener and back again. Annual Reviews in Control, 55:92–117, 2023

work page 2023
[7]

Bilinearization, reachability, and optimal control of control-affine nonlinear systems: A Koopman spectral approach

Debdipta Goswami and Derek A Paley. Bilinearization, reachability, and optimal control of control-affine nonlinear systems: A Koopman spectral approach. IEEE Transactions on Automatic Control, 67(6):2715–2728, 2021

work page 2021
[8]

Nominally robust model predictive control with state constraints

Gene Grimm, Michael J Messina, Sezai E Tuna, and Andrew R Teel. Nominally robust model predictive control with state constraints. IEEE Transactions on Automatic Control, 52(10):1856–1870, 2007

work page 2007

Show all 25 references

[9]

Nonlinear model predictive control

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

2017
[10]

Analysis of unconstrained nonlinear MPC schemes with time varying control horizon

Lars Gr ¨une, J¨urgen Pannek, Martin Seehafer, and Karl Worthmann. Analysis of unconstrained nonlinear MPC schemes with time varying control horizon. SIAM Journal on Control and Optimization, 48(8):4938–4962, 2010

2010
[11]

Koopman form of nonlinear systems with inputs

Lucian Cristian Iacob, Roland T ´oth, and Maarten Schoukens. Koopman form of nonlinear systems with inputs. Auto- matica, 162:111525, 2024. 12 KERNEL EDMD FOR DATA-DRIVEN NONLINEAR KOOPMAN MPC WITH STABILITY GUARANTEES

2024
[12]

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

Frederik K ¨ohne, Friedrich M Philipp, Manuel Schaller, Anton Schiela, and Karl Worthmann. L∞-error bounds for approximations of the Koopman operator by kernel extended dynamic mode decomposition.SIAM J. Applied Dynamical Systems, 2024. To appear (arXiv:2403.18809)

2024 arXiv
[13]

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

Milan Korda and Igor Mezi ´c. Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control. Automatica, 93:149–160, 2018

2018
[14]

A stabilizing model-based predictive control algorithm for nonlinear systems

Lalo Magni, Giuseppe De Nicolao, Lorenza Magnani, and Riccardo Scattolini. A stabilizing model-based predictive control algorithm for nonlinear systems. Automatica, 37(9):1351–1362, 2001

2001
[15]

Introduction to the Koopman Operator in Dynamical Systems and Control Theory, pages 3–33

Alexandre Mauroy, Yoshihiko Susuki, and Igor Mezi ´c. Introduction to the Koopman Operator in Dynamical Systems and Control Theory, pages 3–33. Springer Int. Publishing, Cham, 2020

2020
[16]

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

Feliks N ¨uske, Sebastian Peitz, Friedrich Philipp, Manuel Schaller, and Karl Worthmann. Finite-data error bounds for Koopman-based prediction and control. Journal of Nonlinear Science, 33:14, 2023

2023
[17]

Otto, and Clarence W

Sebastian Peitz, Samuel E. Otto, and Clarence W. Rowley. Data-driven model predictive control using interpolated Koopman generators. SIAM Journal on Applied Dynamical Systems, 19(3):2162–2193, 2020

2020
[18]

Dynamic mode decomposition with control

Joshua L Proctor, Steven L Brunton, and J Nathan Kutz. Dynamic mode decomposition with control. SIAM Journal on Applied Dynamical Systems, 15(1):142–161, 2016

2016
[19]

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

Manuel Schaller, Karl Worthmann, Friedrich Philipp, Sebastian Peitz, and Feliks N ¨uske. Towards reliable data-based optimal and predictive control using extended DMD. IFAC-PapersOnLine, 56(1):169–174, 2023

2023
[20]

Scattered data approximation, volume 17

Holger Wendland. Scattered data approximation, volume 17. Cambridge University Press, 2004

2004
[21]

A note on persistency of excitation

Jan C Willems, Paolo Rapisarda, Ivan Markovsky, and Bart LM De Moor. A note on persistency of excitation. Systems & Control Letters, 54(4):325–329, 2005

2005
[22]

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

Matthew O Williams, Ioannis G Kevrekidis, and Clarence W Rowley. A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition. Journal of Nonlinear Science, 25:1307–1346, 2015

2015
[23]

Estimates on the prediction horizon length in MPC

Karl Worthmann. Estimates on the prediction horizon length in MPC. InProc. 20th Int. Symp. on Mathematical Theory of Networks and Systems (MTNS), Melbourne, Australia, 2012. https://epub.uni-bayreuth.de/id/eprint/ 5657/1/worthmann_mtns_2012.pdf

2012
[24]

Data-driven MPC with termi- nal conditions in the Koopman framework

Karl Worthmann, Robin Str ¨asser, Manuel Schaller, Julian Berberich, and Frank Allg¨ower. Data-driven MPC with termi- nal conditions in the Koopman framework. In 63rd IEEE Conference on Decision and Control (CDC), 2024. To appear

2024
[25]

Learning deep neural network representations for Koopman operators of nonlinear dynamical systems

Enoch Yeung, Soumya Kundu, and Nathan Hodas. Learning deep neural network representations for Koopman operators of nonlinear dynamical systems. In 2019 IEEE American Control Conference (ACC), pages 4832–4839, 2019

2019

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [3]

Data-driven MPC with stability guarantees using extended dynamic mode decomposition

Lea Bold, Lars Gr ¨une, Manuel Schaller, and Karl Worthmann. Data-driven MPC with stability guarantees using extended dynamic mode decomposition. IEEE Transactions on Automatic Control, 70(1):534–541, 2025

work page 2025

[2] [4]

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

Lea Bold, Friedrich M Philipp, Manuel Schaller, and Karl Worthmann. Kernel-based Koopman approximants for control: Flexible sampling, error analysis, and stability. Preprint arXiv:2412.02811, 2024

arXiv 2024

[3] [1]

Non- parametric control-Koopman operator learning: Flexible and scalable models for prediction and control

Petar Bevanda, Bas Driessen, Lucian Cristian Iacob, Roland Toth, Stefan Sosnowski, and Sandra Hirche. Non- parametric control-Koopman operator learning: Flexible and scalable models for prediction and control. Preprint arXiv:2405.07312, 2024

arXiv 2024

[4] [2]

Stability and feasibility of state constrained MPC without stabilizing terminal constraints

Andrea Boccia, Lars Gr ¨une, and Karl Worthmann. Stability and feasibility of state constrained MPC without stabilizing terminal constraints. Systems & Control Letters, 72:14–21, 2014

work page 2014

[5] [5]

Fasshauer and Q

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

work page 2011

[6] [6]

Behavioral theory for stochastic systems? A data-driven journey from Willems to Wiener and back again

Timm Faulwasser, Ruchuan Ou, Guanru Pan, Philipp Schmitz, and Karl Worthmann. Behavioral theory for stochastic systems? A data-driven journey from Willems to Wiener and back again. Annual Reviews in Control, 55:92–117, 2023

work page 2023

[7] [7]

Bilinearization, reachability, and optimal control of control-affine nonlinear systems: A Koopman spectral approach

Debdipta Goswami and Derek A Paley. Bilinearization, reachability, and optimal control of control-affine nonlinear systems: A Koopman spectral approach. IEEE Transactions on Automatic Control, 67(6):2715–2728, 2021

work page 2021

[8] [8]

Nominally robust model predictive control with state constraints

Gene Grimm, Michael J Messina, Sezai E Tuna, and Andrew R Teel. Nominally robust model predictive control with state constraints. IEEE Transactions on Automatic Control, 52(10):1856–1870, 2007

work page 2007