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REVIEW 3 major objections 4 minor 49 references

Koopman-based control of nonlinear systems with closed-loop guarantees

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A data-driven Koopman method returns controllers with provable closed-loop stability and performance guarantees for unknown nonlinear systems.

desk verdict Solid tutorial-plus-extension; the new continuous-time SOS and performance results are plausible and useful, but the advertised end-to-end guarantee is conditional on an unverified projection-error bound. read the letter →

arxiv 2411.10359 v4 pith:DKH24G4Q submitted 2024-11-15 eess.SY cs.SYmath.OC

Robin Strässer , Julian Berberich , Manuel Schaller , Karl Worthmann , Frank Allgöwer This is my paper · ORCID
classification eess.SYcs.SYmath.OC MSC 93C1093D1537N35
keywords KoopmanoperatorextendeddynamicmodedecompositionSafEDMDdata-drivencontrolsum-of-squaresoptimizationclosed-loopguaranteesnonlinearstabilizationL2-gain
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 aims to turn raw state-input data into a feedback controller for an unknown nonlinear system together with a mathematical certificate that the controller works. The approach learns a finite-dimensional bilinear surrogate model of the system's Koopman operator through the SafEDMD procedure, which supplies a guaranteed bound on the model's residual error that is proportional to the lifted state norm and the input norm. Using that bound as uncertainty, sum-of-squares optimization produces a lifted polynomial feedback law that provably renders the origin exponentially stable with a certified convergence rate, and the same construction enforces performance criteria such as a bound on the $\mathcal L_2$-gain from disturbances to user-chosen outputs. If the claims hold, safety-critical systems can be controlled from data alone, with probabilistic but certified closed-loop guarantees rather than heuristic ones.

What carries the argument

The load-bearing construction is the certified bilinear surrogate model (19), $(\sigma\hat\Phi)(x)=A\hat\Phi(x)+B_0u+\tilde B(u\otimes\hat\Phi(x))+r(x,u)$, with the residual confined to the proportional set $\|r(x,u)\|\le c_x\|\hat\Phi(x)\|+c_u\|u\|$ of (21). This particular shape lets a matrix-valued sum-of-squares inequality—condition (22) in continuous time, (24) in discrete time, (34) with performance—imply a genuine Lyapunov inequality for $V(x)=\hat\Phi(x)^\top P^{-1}\hat\Phi(x)$ via the S-procedure, using the multiplier $\tau(\hat\Phi(x))$. The controller $\mu(x)=L(\hat\Phi(x))P^{-1}\hat\Phi(x)$ is the feedback that makes the closed-loop lifted dynamics $A+B_0K(z)+\tilde B(K(z)\otimes z)$ fit into that inequality, and $\Omega(c^*)$ is the largest sublevel set of $V$ inside the sampling region $X$, chosen so that the data-certified error bound remains valid along all trajectories.

What would settle it

Pick a specific dictionary and system, compute the actual residual $r(x,u)$ on a dense grid, and check whether the ratio $\|r(x,u)\vert/(\|\hat\Phi(x)\|+\|u\|)$ stays bounded as $(x,u)\to(0,0)$; if the ratio grows without bound, the proportional bound (21) fails and the SOS certificate loses its validity. Alternatively, simulate the certified controller on the true system from initial conditions inside $\Omega(c^*)$ and look for a trajectory that leaves $\Omega(c^*)$ or fails to converge within the $1-\delta$ probability set, which would directly contradict Theorem 3.

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Extended reading notes

Core claim

The paper's central claim is Theorem 3: whenever the sum-of-squares condition (22) is feasible for the bilinear surrogate model $\dot{\hat\Phi}(x)=A\hat\Phi(x)+B_0u+\tilde B(u\otimes\hat\Phi(x))+r(x,u)$ together with the certified proportional residual bound $\|r(x,u)\le c_x\|\hat\Phi(x)\|+c_u\|u\|$ of Proposition 1, the lifted polynomial feedback $\mu(x)=L(\hat\Phi(x))P^{-1}\hat\Phi(x)$ exponentially stabilizes the origin of the unknown control-affine system (1) for every initial condition in the Lyapunov level set $\Omega(c^*)$ contained in the sampling region $X$, with probability $1-\delta$ over the training data. Theorem 7 and Corollary 9 extend the same machinery to the disturbed system (26), guaranteeing exponential stability together with a quadratic performance inequality, for example a finite $\mathcal L_2$-gain from the disturbance $w$ to the performance output $y$. The discrete-time counterpart, Theorem 4, obtains the same stability guarantee with a rational sampled-data controller and needs only state-successor measurements rather than state derivatives. What makes these statements non-vacuous is the proportional shape of the residual bound: because the bound vanishes at the origin, the Lyapunov argument closes at $(x,u)=(0,0)$, so the certificate covers a neighborhood of the equilibrium rather than only a compact set.

Load-bearing premise

The main load-bearing premise is that the surrogate's residual stays below a known bound that scales linearly with the lifted state and the input, with concrete constants; the paper notes that deriving such proportional bounds for the projection error in practice is an open problem, and the numerical examples choose those constants by hand rather than computing them from data.

Editorial extensions

If this is right

  • A user who collects the required state-input samples can obtain, for any confidence $1-\delta$, a feedback law with a certified exponential convergence rate and a certified region of attraction, without ever identifying the drift or input maps $f$ and $g_i$.
  • The SOS-based design tolerates larger residual bounds than the earlier LMI-based SafEDMD controllers, so it remains feasible with fewer data samples and returns a larger guaranteed region of attraction (Section 6, Figure 1).
  • By choosing performance output matrices, the same SOS program certifies a quadratic supply-rate inequality—e.g., a guaranteed $\mathcal L_2$-gain bound—for the closed-loop nonlinear system, provided trajectories stay inside the sampling region.
  • In discrete time, the sampled-data controller needs no derivative measurements, only state-successor pairs, and still guarantees exponential stability of the continuous-time system between sampling instants.
  • The learning-error part of the bound shrinks as $\mathcal O(1/\sqrt{\delta d})$ (plus $\Delta t^2$ in discrete time), so the guarantees can be sharpened by collecting more data with a small sampling period.

Reading between the lines

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

  • If the open problem of deriving practical proportional projection bounds is resolved—the paper points to kernel-EDMD results—the framework becomes a fully automatic data-to-certificate pipeline with no hand-chosen error constants.
  • The robust invariant set $\Omega(c^*)$ produced by the certificate could be reused as a terminal set or constraint set in model predictive control, connecting this design to the MPC extensions the authors mention.
  • A natural stress test is to benchmark the certified $\mathcal L_2$-gain against the true nonlinear gain on systems with known finite-dimensional Koopman invariant subspaces; the gap would quantify the remaining conservatism of the SOS approach.
  • The discrete-time performance extension is left for future work; combining the rational controller parametrization of Theorem 4 with the performance matrix (34) would plausibly yield sampled-data $\mathcal L_2$-gain certificates.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper presents a tutorial overview and extension of the SafEDMD framework for data-driven control of unknown control-affine nonlinear systems. A bilinear surrogate model of the lifted Koopman dynamics is estimated from data, and a proportional residual bound of the form ||r(x,u)|| <= c_x ||Phi_hat(x)|| + c_u ||u|| is used to synthesize feedback controllers via sum-of-squares (SOS) optimization. The authors prove exponential stabilization guarantees in continuous and discrete time (Theorems 3 and 4) and extend the approach to quadratic performance specifications such as an L2-gain bound (Theorem 7 and Corollary 9). A numerical example compares the proposed SOS-based controllers with earlier LMI-based designs and shows larger guaranteed regions of attraction and feasible uncertainty constants.

Significance. If the key proportional error bound can be certified from data, the framework would provide a rigorous and practical pipeline from raw data to closed-loop stability and performance certificates for unknown nonlinear systems. The paper's explicit contribution is the continuous-time SOS controller design and the performance-oriented extension, which improves on the earlier LMI-based SafEDMD designs. The paper is transparent about its assumptions and provides proofs in the appendices, and the authors share code for the numerical experiments. However, the advertised end-to-end guarantee is currently conditional on an unverified assumption on the dictionary projection error, which substantially limits the strength of the claims. The numerical experiments do not close this gap because the uncertainty constants are chosen by hand rather than computed or validated.

major comments (3)
  1. [Section 3.3, Proposition 1 and Eq. (21)] The load-bearing proportional projection-error bound with constants c~_x and c~_u is assumed, not derived. The paragraph immediately after the proof of Proposition 1 states that deriving rigorous proportional bounds on the projection error is 'an interesting and challenging problem for future research.' Since c_x = c~_x + \bar c_x and c_u = c~_u + \bar c_u enter every SOS condition (22), (24), and (34), the closed-loop stability and performance guarantees are conditional on an unverified assumption about the chosen dictionary. The paper should either provide a method to compute or certify c~_x and c~_u for a concrete class of dictionaries, or state the main theorems explicitly as conditional on a verifiable certificate and indicate how that certificate would be obtained.
  2. [Section 6, Eq. (37) and Figs. 1-2] The numerical experiments do not validate the key assumption behind the guarantees. The constants c_x = c_u are chosen in (37) as the largest values for which the comparison LMI controllers from [30,31] are feasible, rather than computed from the dictionary (36), the d = 200 data points, or the sampling period \Delta t = 0.01. No evidence is shown that the actual residual r(x,u) of the SafEDMD surrogate satisfies (21) with these constants on X = [-1,1]^2 with probability 0.95. Consequently, the guaranteed regions of attraction in Fig. 1 and the guaranteed L2-gain curve in Fig. 2 are certificates for a hypothetical uncertainty set, not for the simulated nonlinear system (35). The authors should compute the required constants from the error analysis or provide a rigorous finite-sample verification of the proportional bound (e.g., by bounding the worst-case ratio ||r(x,u)||/(||Phi_hat(x)|| + ||u||) over X and the input range).
  3. [Section 5, Theorem 7 and Corollary 9] Theorem 7 assumes that the proportional error bound (21) holds for all x in R^n, whereas Proposition 1 only provides such a bound on a compact sampling set X with probability 1 - delta. Corollary 9 is the statement whose hypotheses match the SafEDMD setting, but the paper presents Theorem 7 as the main performance result and only later qualifies it. This presentation can mislead readers into thinking that global performance guarantees are available from the data-driven framework. The paper should restructure Section 5 so that the local theorem (Corollary 9) is the main data-driven performance statement, with Theorem 7 clearly labeled as an idealized auxiliary result whose global hypothesis must be verified separately.
minor comments (4)
  1. [Section 6, Remark 10 and dictionary (36)] The dictionary (36) contains the term x_2 - lambda/(lambda - 2 rho) x_1^2, which uses exact knowledge of the system parameters lambda and rho. Since the system is assumed unknown, this choice cannot be made in a purely data-driven procedure. The remark about the example's tutorial value does not remove the concern that the numerical demonstration is partly model-informed.
  2. [Section 4.2, Theorem 4] Theorem 4 is quoted from [34] without a proof in this paper. For a tutorial-style contribution this is acceptable, but the paper should state more explicitly which parts are new and which are restatements of prior work, so that the reader can distinguish the survey content from the original contributions.
  3. [Section 6, Fig. 1] The comparison between continuous-time and discrete-time SOS controllers in Fig. 1 uses different uncertainty constants (0.1 vs. 0.006), so the comparison of the regions of attraction is not apples-to-apples. This should be stated in the text or the figure caption.
  4. [After Proposition 1] The paper cites [39] as showing that a nonlinear data-informed dictionary admits a proportional error bound. Since this is directly relevant to the main assumption, the authors should state the concrete class of dictionaries and the resulting constants, or at least include the relevant result in a remark, so that the reader can see how the assumption can be satisfied in practice.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the SOS stability and performance theorems are derived from the stated assumptions; the projection-error bound in Proposition 1 is an explicit assumption rather than a conclusion extracted from data.

full rationale

Walking the derivation chain: Section 3.3 defines the residual model (19) and Proposition 1 states a proportional bound (21) whose constants are c_x = cbar_x + ctilde_x. The learning-error part cbar_x, cbar_u is cited to [30,31]; the projection-error part ctilde_x, ctilde_u is explicitly assumed, with the paper stating that deriving such bounds is an open problem. The controller theorems (Theorem 3, Theorem 7, Corollary 9) are then proved in Appendices A-C as implications of that assumption: if (21) holds, the SOS conditions (22)/(34) imply the Lyapunov inequality and quadratic performance. This is a conditional guarantee, not a circular one, because the theorems do not assert that (21) has been verified for the actual data. The numerical section sets c_x=c_u=0.1 and 0.006 as the maximal values for which comparison LMI controllers are feasible, rather than computing them from data; this is a verification gap or over-claim, not an instance of a predicted quantity being equal to a fitted input by construction. The citations to the authors' prior SafEDMD work [30,31,34] are load-bearing for the error-bound and discrete-time results, but they are published proofs with stated assumptions, so they count as independent support under the review rules. No step reduces to its own input by definition; hence the circularity score is low.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the SafEDMD error certification from the authors' earlier papers and on an assumed proportional projection-error bound with unspecified constants. The dictionary choice and the robust-control SOS formulation are standard. No new physical or mathematical entities are invented.

free parameters (3)
  • projection error constants c~_x and c~_u
    Assumed to exist in Proposition 1; no method is provided to compute them, and they directly set c_x and c_u in the uncertainty bound (21).
  • c_x = c_u in continuous-time numerical example = 0.1
    Chosen as the maximal values for which the LMI controller from [30] is feasible; not estimated from the d=200 samples.
  • c_x = c_u in discrete-time numerical example = 0.006
    Chosen as the maximal values for which the LMI controller from [31] is feasible; not estimated from data.
assumptions (4)
  • ad hoc to paper The projection error of the dictionary satisfies a proportional bound |(L^u - P_V L^u|_V)Phi(x)| <= c~_x |Phi_hat(x)| + c~_u |u| (or its discrete-time analogue) with known constants.
    Assumed in Proposition 1; the following paragraph states that deriving such bounds is future research, so the end-to-end guarantee depends on this unproved premise.
  • domain assumption Data samples are drawn i.i.d. from a compact set X.
    Used in Proposition 1 to get the O(1/sqrt(delta d)) plus O(Delta t^2) probabilistic learning-error bounds.
  • standard math The Koopman generator is control affine: L^u = L^0 + sum_i u_i (L^{e_i} - L^0).
    Used to derive the bilinear surrogate models (13) and (17) from the data-driven matrices.
  • domain assumption The dictionary satisfies Phi(0)=0 and |x| <= |Phi_hat(x)| <= L_Phi |x| on X (Eq. (6)).
    Needed to translate decay of the lifted state into exponential stability of the original state and to define the region Omega(c*).

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Cite this review

Pith. "Pith review of Koopman-based control of nonlinear systems with closed-loop guarantees." pith.science (2026). https://pith.science/paper/DKH24G4Q

@misc{pith2026241110359,
  author       = {Pith},
  title        = {Pith review of: Koopman-based control of nonlinear systems with closed-loop guarantees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DKH24G4Q}},
  note         = {Machine review of arXiv:2411.10359}
}
read the original abstract

In this paper, we provide a tutorial overview and an extension of a recently developed framework for data-driven control of unknown nonlinear systems with rigorous closed-loop guarantees. The proposed approach relies on the Koopman operator representation of the nonlinear system, for which a bilinear surrogate model is estimated based on data. In contrast to existing Koopman-based estimation procedures, we state guaranteed bounds on the approximation error using the stability- and certificate-oriented extended dynamic mode decomposition (SafEDMD) framework. The resulting surrogate model and the uncertainty bounds allow us to design controllers via robust control theory and sum-of-squares optimization, guaranteeing desirable properties for the closed-loop system. We present results on stabilization both in discrete and continuous time, and we derive a method for controller design with performance objectives. The benefits of the presented framework over established approaches are demonstrated with a numerical example.

Figures

Figures reproduced from arXiv: 2411.10359 by the authors.

Figure 1
Figure 1. Sampling region X ( ), RoA for the continuous￾time surrogate model with SOS controller ( ) and LMI con￾troller ( ), and RoA for the discrete-time surrogate model with SOS controller ( ) and LMI controller ( ). We note that the constants in (37) are the maximal values for which the LMI-based controllers [30, 31], to which we compare our proposed SOS-based controller, are feasible. Note that the maximal feasible value… view at source ↗
Figure 2
Figure 2. Guaranteed ℒ2-gain bound 𝛾 of Corollary 9 for different values of 𝑐𝑥 = 𝑐𝑢 in the proportional error bound (21). increasing uncertainty in the surrogate model deterio￾rates the guaranteed closed-loop performance. 7 Conclusion We presented a framework for designing data-driven controllers for unknown nonlinear systems via the Koopman operator. The proposed approach relied on the SafEDMD framework, which provides a bil… view at source ↗

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.