Pith. sign in

REVIEW 4 major objections 4 minor 40 references

Koopman Meets Limited Bandwidth: Effect of Quantization on Data-Driven Linear Prediction and Control of Nonlinear Systems

T0 review · 4 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 Dither-quantized training data turn Koopman linear-predictor identification into a regularized least-squares problem, with a penalty of order the square of the quantization step.

desk verdict Solid incremental extension of the authors' own quantization–EDMD work; the large-data regularization theorem is plausible but not actually validated by the paper's own non-analytic dictionaries. read the letter →

arxiv 2501.07714 v1 pith:VCYH3EUE submitted 2025-01-13 eess.SY cs.SY

Shahab Ataei , Dipankar Maity , Debdipta Goswami This is my paper · ORCID
classification eess.SYcs.SY MSC 93B3093C10
keywords KoopmanoperatorExtendedDynamicModeDecompositionditherquantizationsystemidentificationmodelpredictivecontrolliftedlinearpredictorregularizedleastsquaresresolution
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

Sending state and control snapshots through a dither quantizer before Koopman-based identification is not just adding noise: this paper claims it is equivalent, in the large-data limit, to solving a regularized least-squares problem. The regularization penalty grows as the square of the quantization step $\epsilon$, through two matrices $\beta(\epsilon)$ and $\Gamma(\epsilon)$ built from power series coefficients of the squared prediction residual, so the estimate from quantized data converges almost surely to a shrunk version of the unquantized estimate. In the finite-data regime the estimate differs from the unquantized one by $O(\epsilon)$, which predicts the linear log-error versus word-length slope seen in the experiments. If true, this gives practitioners a rule for how many bits are needed to keep a prescribed identification error, and it explains why moderate quantization can even improve downstream model predictive control performance by acting as a regularizer.

What carries the argument

The load-bearing object is the one-step squared residual $r(x_{t+1},x_t,u_t)=\|\varphi(x_{t+1})-A\varphi(x_t)-Bu_t\|^2$, viewed as a function of the dither-quantized snapshots. Dither quantization makes each component of the error $\tilde{x}-x$ and $\tilde{u}-u$ i.i.d. uniform on $[-\epsilon/2,\epsilon/2]$, so in the power series expansion of $r$ around the noisy snapshot, odd-order terms vanish almost surely under the strong law of large numbers and even-order terms converge to deterministic traces involving second derivatives of $\varphi$. These traces assemble into the matrices $\beta(\epsilon)$ and $\Gamma(\epsilon)$ that produce the regularization. The finite-$T$ bound uses the mean-value theorem plus the gradient bound $\|\nabla\varphi_i\|\le c_\varphi$ to show $\|\Phi_\epsilon\|=O(\epsilon)$ and hence $\|\tilde{G}-G\|=O(\epsilon)$.

What would settle it

Run modified EDMD on a controlled system whose dictionary contains a thin-plate spline centered on a training trajectory, with very large $T$ and fixed $\epsilon$; compute the almost-sure limit of $[\tilde{A},\tilde{B}]$ and compare it with the minimizer of the unquantized objective plus $\operatorname{tr}(G\beta(\epsilon))+\operatorname{tr}(G^{\top}G\Gamma(\epsilon))$ evaluated by finite-difference second-derivative matrices. If the limits do not match, Theorem 1 fails exactly where Assumption 3 is violated.

Watch

Extended reading notes

Core claim

The central claim is a limit identity. Let $G=[A,B]$ be the lifted linear predictor obtained by modified EDMD from unquantized snapshots, and $[\tilde{A},\tilde{B}]$ the predictor from snapshots that have been dithered and quantized with resolution $\epsilon$. Under assumptions that make the one-step residual $r(x_{t+1},x_t,u_t)=\|\varphi(x_{t+1})-A\varphi(x_t)-Bu_t\|^2$ analytic in a neighborhood of the quantized data, Theorem 1 states that as $T\to\infty$ the quantized estimate converges almost surely to the minimizer of $\limsup_{T\to\infty}\frac{1}{T}\|\Phi_{+}-A\Phi-BU\|^2+\operatorname{tr}(G\beta(\epsilon))+\operatorname{tr}(G^{\top}G\Gamma(\epsilon))$, with $\beta,\Gamma=O(\epsilon^2)$. Theorem 2 states that for finite full-rank data the difference is $G_\epsilon$ with $\|G_\epsilon\|=O(\epsilon)$. When the observables themselves are quantized rather than the raw states, the regularizer becomes the explicit ridge penalty $\frac{\epsilon^2}{12}(\|A\|^2+\|B\|^2)$, and under a data-growth condition on the snapshot covariance the normalized identification error can be driven to zero for any fixed $\epsilon$.

Load-bearing premise

The large-data theorem requires that the squared prediction residual, as a function of the measured snapshot, can be expanded in a convergent power series around every point the quantizer can produce, with radius bigger than half the quantization step; this is violated by the thin-plate-spline radial basis functions used in the paper's experiments, whose $r^2\log r$ form is not analytic at its center.

Editorial extensions

If this is right

  • Designers on bandwidth-constrained hardware can budget bits by the rule that asymptotic bias scales as $\epsilon^2 = \left((x_{\max}-x_{\min})/2^b\right)^2$: every extra bit quarters the regularization-induced bias.
  • Finite-data identification error is $O(\epsilon)$, so log-relative-error falls with slope about $-\log 2$ per bit; the reported experiments show this slope.
  • Quantizing observables directly rather than states removes all dictionary dependence from the regularizer, yielding $(\epsilon^2/12)(\|A\|^2+\|B\|^2)$, and Lemma 2 says enough data can wash out quantization entirely in that setting.
  • The unquantized predictor is recoverable in principle by solving the debiased problem (20), so the theory points toward a correction algorithm once $\beta$ and $\Gamma$ are estimated.
  • MPC with quantized-data predictors approaches unquantized performance as word-length grows; non-monotone cases such as one oscillator at $b=8$ are consistent with beneficial regularization rather than pure error.

Reading between the lines

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

  • If the asymptotic equivalence holds generally, quantization resolution becomes a regularization dial: deliberately dithering and quantizing before EDMD could improve generalization in small-sample regimes, and one could tune word-length as a hyperparameter just like a ridge penalty.
  • Because $\beta$ and $\Gamma$ depend on second derivatives of the dictionary, quantization's effect is dictionary-dependent; under a fixed bit budget, observables with small curvature over the data distribution should be preferred to minimize induced bias.
  • A concrete testable extension is to estimate $\beta(\epsilon)$ and $\Gamma(\epsilon)$ by finite differences of the residual second-derivative matrices on the quantized snapshots and then apply (20); success would turn the asymptotic equivalence into a practical debiasing tool.
  • The paper's own thin-plate-spline experiments sit outside Assumption 3 because $r^2\log r$ is not analytic at its center; if convergence still holds there, the true sufficient condition is likely a weaker smoothness or a decay of high-order terms, which would be worth isolating.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies the effect of dither quantization on the EDMD-based lifted linear predictor for nonlinear control systems. The state and input snapshots are quantized before lifting, and the least-squares estimate of the predictor matrices [A,B] is compared with the unquantized estimate. The central theoretical claims are: (1) in the large-data limit T→∞, the quantized estimate converges almost surely to the minimizer of the unquantized quadratic objective plus an O(ε²) regularization term of the form tr(Gβ(ε))+tr(GᵀGΓ(ε)) (Theorem 1); (2) for finite T, the difference between the quantized and unquantized estimates is O(ε) (Theorem 2); and (3) when the observables themselves are quantized, the limiting objective is exactly the unquantized objective plus (ε²/12)(‖A‖²+‖B‖²), giving an explicit regularized least-squares recovery rule (Theorem 3). The paper also reports numerical experiments on a negatively damped pendulum, a Van der Pol oscillator, a bilinear DC motor, and a KdV PDE, with a Koopman-based linear MPC tracking problem, showing error scaling and cost degradation with decreasing word length.

Significance. If the main results are correct, the paper makes a useful conceptual contribution: dither quantization in EDMD is not merely unstructured noise but acts as a deterministic, dictionary-dependent implicit regularization in the large-data limit, with an explicit O(ε²) scaling that connects quantization resolution to prediction/control performance. This is potentially valuable for networked and resource-limited Koopman-based control. The paper is self-contained in its derivations and does not rely on black-box external results for the central theorems; it uses standard tools (Schuchman's condition, strong laws of large numbers) and provides an interesting special case in Section VI with an explicit closed-form regularity term. The experimental validation covers four different systems and includes both prediction and MPC, which is commendable. However, the stated Theorem 1 excludes the thin-plate-spline dictionaries used in two of the four experiments, and the proofs of Theorems 1 and 2 have gaps in the interchange of limits, uniform convergence, and control of inverse matrices.

major comments (4)
  1. [§V-A, §V-B and §IV-A, Assumption 3] The pendulum and Van der Pol experiments use a dictionary containing 100 thin-plate-spline RBFs ψ(x)=‖x−c‖²log‖x−c‖. This function is not C² at its center c and has no Taylor expansion at c. Since the centers are drawn uniformly in the unit box and the trajectories explore the same box, for any fixed ε the data will, with high probability, contain snapshots within distance ε/2 of a center as T grows; for the reported finite T with 200,000 snapshots and 100 centers this is already likely. Consequently Assumption 3, which requires the residual r to have an absolutely convergent Taylor series with radius of convergence larger than ε/2, is violated in exactly the experiments that are used to validate Theorem 1. The paper should either replace the thin-plate-spline dictionary with an analytic dictionary (e.g., Gaussian RBFs) or weaken Assumption 3 and rework Theorem 1 to cover non-smooth observables; as it stands, two of the four case studies cannot be cited as numerical support for the large-data theorem.
  2. [Appendix B, Eq. (50)–(52)] The proof of Theorem 1 expands the residual in an infinite Taylor series and then interchanges the limit T→∞ with the infinite sum. Assumption 3 only states pointwise absolute convergence of the Taylor series for each fixed data point; it does not provide the uniform domination over t, over G, or over the tail k>n that is needed to interchange the order of the T-limit and the n-limit. Furthermore, Eq. (52) passes the argmin through the almost-sure limit of the objective functions without a uniform-convergence argument over the parameter set A×B. These are load-bearing steps for the paper's central result. The argument can likely be repaired with additional boundedness and domination assumptions (or by proving epi-convergence/consistency of the minimizers), but as written the proof of Theorem 1 is incomplete.
  3. [Theorem 2 proof, Eq. (22)–(23)] The finite-data proof writes ̅G = G − G(ΨΨᵀΞ_ε⁻¹ + I)⁻¹ + Π_ε(̅Ψ̅Ψᵀ)⁻¹ and then concludes ∥G_ε∥=O(ε) from ∥Ξ_ε∥=O(ε) and ∥Π_ε∥=O(ε). This conclusion requires uniform control of the inverses: the term (ΨΨᵀΞ_ε⁻¹ + I)⁻¹ contains Ξ_ε⁻¹, whose norm may grow like 1/ε, and the full-row-rank assumption on ̅Ψ does not by itself give a lower bound on the smallest singular value of ̅Ψ̅Ψᵀ that is uniform as ε→0. Without explicit bounds of the form ∥(ΨΨᵀΞ_ε⁻¹+I)⁻¹∥≤C and ∥(̅Ψ̅Ψᵀ)⁻¹∥≤C/λ_min, the claimed O(ε) estimate is not established. This is a essential gap in Theorem 2, which is the basis for the finite-data error scaling used throughout the experimental section.
  4. [Appendix A, Corollaries 3–4] Corollary 3 states that E[y_k(y_ℓ)³]=0 follows from pairwise uncorrelatedness of {y_t}; this implication is false in general, and the proof needs to use the specific independence structure of products of disjoint and overlapping quantization errors. Additionally, Corollary 4 invokes Kolmogorov's strong law (Lemma 5) for sequences a_t e_t; for the law to apply, the e_t must be independent of the coefficients a_t, not merely mean-zero and uncorrelated. The paper should explicitly assume that the dither noises are independent of the underlying trajectory and control sequence and then state the independence of the product sequence. As written, the applications of the strong law in Appendix B are not fully justified, although they are likely repairable.
minor comments (4)
  1. [§II-C] In Lemma 1, the phrase "characteristic equation" should be "characteristic function".
  2. [Appendix A, Corollary 3 proof] In the fourth-moment computation, the text says "the second inequality follows" but the step is an equality; the wording should be corrected.
  3. [Appendix C, definitions of ̃Ψ and ̅U] The definition of ̅U in the proof of Theorem 3 is inconsistent with the definition in Section III: it lists ũ1,...,ũ_{T−1} and omits the term ũ_T that appears in Φ+. The indexing should be harmonized.
  4. [General] The paper would benefit from error bars or confidence intervals in the experimental figures; with 50 Monte Carlo realizations, the slopes and non-monotonicities discussed in Section V-B could be assessed more reliably.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorems are derived in-paper from the dither quantization model via Taylor expansion and strong laws of large numbers; the authors' self-citations are framing only.

full rationale

The paper's central claim, Theorem 1, is that the dither-quantized EDMD estimate converges almost surely to the unquantized least-squares objective plus tr(Gβ(ε)) + tr(G'GΓ(ε)). This is not circular: β and Γ are not fitted parameters or renamed outputs; they are constructed in the proof (Appendix B) by expanding r(x_{t+1}+e^x_{t+1}, ξ_t+η_t) in a Taylor series, applying the dither-noise moment identities (Lemma 3 and Corollaries 1–3), and invoking Kolmogorov's strong law of large numbers (Lemma 5). The regularization term emerges from the algebra of the squared residual under dither quantization, so it is a derived consequence, not an assumption equivalent to the result. Theorem 2 similarly derives the O(ε) finite-data bound from the closed-form least-squares solution and the mean-value theorem, with no fitted quantity renamed as a prediction. Theorem 3 repeats the same pattern for quantized observables and produces the explicit ε²/12 (‖A‖² + ‖B‖²) term from an i.i.d. observable-quantization error model. The authors' prior works [18] and [19] are cited as background and as prior efforts on quantized DMD/EDMD; no load-bearing argument reduces to those citations. The only external facts used are standard: Schuchman's condition from [17] and Kolmogorov's SLLN from [39]. The skeptic's concern that Assumption 3 (absolute Taylor convergence of the residual) is violated by the thin-plate-spline dictionaries used in the pendulum and Van der Pol experiments is a correctness/assumption-coverage issue, not a circularity issue: it questions whether Theorem 1 applies to those numerical cases, but it does not show that any prediction is equivalent to its inputs by construction. Likewise, the empirical slope comparisons in Section V are consistency checks, not fitted parameters used to produce the theorems. I therefore find no circular step in the derivation chain and assign a score of 0.

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

The analysis depends on standard dither-quantization statistics and on the paper's assumptions on the lifting functions. No new physical entities are introduced. The most fragile input is the analyticity assumption on the residual, which is not satisfied by the paper's own dictionary choices.

assumptions (6)
  • domain assumption Dither quantization with uniform noise in [-ϵ/2, ϵ/2] yields quantization errors that are i.i.d., uniform in [-ϵ/2, ϵ/2], and independent of the signal (Schuchman's condition).
    Used throughout the proofs (Lemma 3, Theorem 1, Theorem 3); this is why the noise has known moments and zero cross-correlation. It is standard for dithered quantizers.
  • domain assumption Assumption 1: observables and control inputs are bounded.
    Section IV-A; guarantees data matrices bounded and enables LLN applications.
  • domain assumption Assumption 2: there exist sets A and B and c_r>0 such that r < c_r for all candidate predictors.
    Section IV-A; ensures the unquantized EDMD objective is well-posed in the large-data limit.
  • ad hoc to paper Assumption 3: the residual has an absolutely convergent Taylor series with radius of convergence > ϵ/2.
    Section IV-A; used in the proof of Theorem 1 to expand the residual at the noisy snapshots and interchange limits. Violated by the thin plate spline observables used in the experiments.
  • domain assumption Assumption 4: ||∇φ_i(x)|| ≤ c_φ.
    Section IV-A; used to bound the perturbation matrices Φ_ϵ in Theorem 2 and to control the first-order terms in Theorem 1.
  • domain assumption Full row rank of the data matrices Ψ and Ψ̄.
    Section IV-C; required for the closed-form least-squares expressions in Theorem 2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Koopman Meets Limited Bandwidth: Effect of Quantization on Data-Driven Linear Prediction and Control of Nonlinear Systems." pith.science (2026). https://pith.science/paper/VCYH3EUE

@misc{pith2026250107714,
  author       = {Pith},
  title        = {Pith review of: Koopman Meets Limited Bandwidth: Effect of Quantization on Data-Driven Linear Prediction and Control of Nonlinear Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VCYH3EUE}},
  note         = {Machine review of arXiv:2501.07714}
}
read the original abstract

Koopman-based lifted linear identification have been widely used for data-driven prediction and model predictive control (MPC) of nonlinear systems. It has found applications in flow-control, soft robotics, and unmanned aerial vehicles (UAV). For autonomous systems, this system identification method works by embedding the nonlinear system in a higher-dimensional linear space and computing a finite-dimensional approximation of the corresponding Koopman operator with the Extended Dynamic Mode Decomposition (EDMD) algorithm. EDMD is a data-driven algorithm that estimates an approximate linear system by lifting the state data-snapshots via nonlinear dictionary functions. For control systems, EDMD is further modified to utilize both state and control data-snapshots to estimate a lifted linear predictor with control input. This article investigates how the estimation process is affected when the data is quantized. Specifically, we examine the fundamental connection between estimates of the linear predictor matrices obtained from unquantized data and those from quantized data via modified EDMD. Furthermore, using the law of large numbers, we demonstrate that, under a large data regime, the quantized estimate can be considered a regularized version of the unquantized estimate. We also explore the relationship between the two estimates in the finite data regime. We further analyze the effect of nonlinear lifting functions on this regularization due to quantization. The theory is validated through repeated numerical experiments conducted on several control systems. The effect of quantization on the MPC performance is also demonstrated.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 34 canonical work pages

  1. [1]

    Hamiltonian systems and transformation in Hilbert space,

    B. O. Koopman, “Hamiltonian systems and transformation in Hilbert space,” Proceedings of National Academy of Sciences , vol. 17, pp. 315– 318, 1931

    work page 1931

  2. [2]

    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

  3. [3]

    Koopman autoencoder ar- chitecture for current density modeling in kinetic plasma simulations,

    I. Nayak, F. L. Teixeira, and M. Kumar, “Koopman autoencoder ar- chitecture for current density modeling in kinetic plasma simulations,” in 2021 International Applied Computational Electromagnetics Society Symposium (ACES), pp. 1–3, IEEE, 2021

    work page 2021

  4. [4]

    Se(3) koopman-mpc: Data-driven learning and control of quadrotor uavs,

    S. S. K. S. Narayanan, D. Tellez-Castro, S. Sutavani, and U. Vaidya, “Se(3) koopman-mpc: Data-driven learning and control of quadrotor uavs,” 2023

    work page 2023

  5. [5]

    Data-driven analysis and forecasting of highway tra ffic dynamics,

    A. M. Avila and I. Mezi ´c, “Data-driven analysis and forecasting of highway tra ffic dynamics,” Nature Communications , vol. 11, no. 1, p. 2090, 2020

    work page 2020

  6. [6]

    Optimizing neural networks via koopman operator theory,

    A. S. Dogra and W. Redman, “Optimizing neural networks via koopman operator theory,” in Advances in Neural Information Processing Systems (H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin, eds.), vol. 33, pp. 2087–2097, Curran Associates, Inc., 2020

    work page 2020

  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, Dec 2015

    work page 2015

  8. [8]

    Linear predictors for nonlinear dynamical sys- tems: Koopman operator meets model predictive control,

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

    2018

Show all 40 references
  1. [9]

    Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the koopman operator,

    H. Arbabi and I. Mezi ´c, “Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the koopman operator,” SIAM Journal on Applied Dynamical Systems , vol. 16, no. 4, pp. 2096–2126, 2017

    2017

  2. [10]

    Centering data improves the dynamic mode decomposition,

    S. M. Hirsh, K. D. Harris, J. N. Kutz, and B. W. Brunton, “Centering data improves the dynamic mode decomposition,” SIAM Journal on Applied Dynamical Systems , vol. 19, no. 3, pp. 1920–1955, 2020

    1920

  3. [11]

    E ffects of spatial resolution of the data on dynamic mode decomposition and its quantitative analysis,

    J. Lee, L. Zimmer, T. Saito, S. Nakaya, and M. Tsue, “E ffects of spatial resolution of the data on dynamic mode decomposition and its quantitative analysis,” Experimental Thermal and Fluid Science , vol. 151, p. 111077, 2024

    2024

  4. [12]

    Koopnet: Joint learning of koopman bilinear models and function dictionaries with application to quadrotor trajectory tracking,

    C. Folkestad, S. X. Wei, and J. W. Burdick, “Koopnet: Joint learning of koopman bilinear models and function dictionaries with application to quadrotor trajectory tracking,” in 2022 International Conference on Robotics and Automation (ICRA) , pp. 1344–1350, 2022

    2022

  5. [13]

    Machine learning on small uavs,

    A. Cleary, K. Yoo, P. Samuel, S. George, F. Sun, and S. A. Israel, “Machine learning on small uavs,” in 2020 IEEE Applied Imagery Pattern Recognition Workshop (AIPR) , pp. 1–5, 2020

    2020

  6. [14]

    Stabilizability of stochastic linear systems with finite feedback data rates,

    G. N. Nair and R. J. Evans, “Stabilizability of stochastic linear systems with finite feedback data rates,” SIAM Journal on Control and Optimiza- tion, vol. 43, no. 2, pp. 413–436, 2004

    2004

  7. [15]

    Optimal controller synthesis and dynamic quantizer switching for linear-quadratic-Gaussian systems,

    D. Maity and P. Tsiotras, “Optimal controller synthesis and dynamic quantizer switching for linear-quadratic-Gaussian systems,” IEEE Trans- actions on Automatic Control , vol. 67, no. 1, pp. 382–389, 2021

    2021

  8. [16]

    Optimal quantizer scheduling and controller synthesis for partially observable linear systems,

    D. Maity and P. Tsiotras, “Optimal quantizer scheduling and controller synthesis for partially observable linear systems,” SIAM Journal on Control and Optimization , vol. 61, no. 4, pp. 2682–2707, 2023

    2023

  9. [17]

    Dithered quantizers,

    R. M. Gray and T. G. Stockham, “Dithered quantizers,” IEEE Transac- tions on Information Theory , vol. 39, no. 3, pp. 805–812, 1993

    1993

  10. [18]

    On the e ffect of quantization on dynamic mode decomposition,

    D. Maity, D. Goswami, and S. Narayanan, “On the e ffect of quantization on dynamic mode decomposition,” arXiv preprint arXiv:2404.02014 , 2024

    2024 arXiv

  11. [19]

    On the e ffect of quantization on extended dynamic mode decomposition,

    D. Maity and D. Goswami, “On the e ffect of quantization on extended dynamic mode decomposition,” arXiv preprint arXiv:2410.02803 , 2024

    2024 arXiv

  12. [20]

    Dynamic mode decomposition of numerical and experi- mental data,

    P. J. Schmid, “Dynamic mode decomposition of numerical and experi- mental data,” Journal of Fluid Mechanics , vol. 656, pp. 5–28, 2010

    2010

  13. [21]

    Linearly recurrent autoencoder networks for learning dynamics,

    S. E. Otto and C. W. Rowley, “Linearly recurrent autoencoder networks for learning dynamics,” SIAM Journal on Applied Dynamical Systems , vol. 18, no. 1, pp. 558–593, 2019

    2019

  14. [22]

    Learning deep neural network representations for koopman operators of nonlinear dynamical systems,

    E. Yeung, S. Kundu, and N. Hodas, “Learning deep neural network representations for koopman operators of nonlinear dynamical systems,” in 2019 American Control Conference (ACC) , pp. 4832–4839, 2019

    2019

  15. [23]

    Bilinearization, reachability, and op- timal control of control-a ffine nonlinear systems: A koopman spectral approach,

    D. Goswami and D. A. Paley, “Bilinearization, reachability, and op- timal control of control-a ffine nonlinear systems: A koopman spectral approach,” IEEE Transactions on Automatic Control , vol. 67, no. 6, pp. 2715–2728, 2022

    2022

  16. [24]

    Data-driven model predictive control using interpolated Koopman generators,

    S. Peitz, S. E. Otto, and C. W. Rowley, “Data-driven model predictive control using interpolated Koopman generators,” SIAM Journal on Applied Dynamical Systems , vol. 19, no. 3, pp. 2162–2193, 2020

    2020

  17. [25]

    Dither signals and their e ffect on quantization noise,

    L. Schuchman, “Dither signals and their e ffect on quantization noise,” IEEE Transactions on Communication Technology , vol. 12, no. 4, pp. 162–165, 1964

    1964

  18. [26]

    Quantization and dither: A theoretical survey,

    S. P. Lipshitz, R. A. Wannamaker, and J. Vanderkooy, “Quantization and dither: A theoretical survey,” Journal of the audio engineering society , vol. 40, no. 5, pp. 355–375, 1992

    1992

  19. [27]

    Picture coding using pseudo-random noise,

    L. Roberts, “Picture coding using pseudo-random noise,” IRE Transac- tions on Information Theory , vol. 8, no. 2, pp. 145–154, 1962

    1962

  20. [28]

    Structural stabilization and quenching by dither in nonlinear systems,

    G. Zames and N. Shneydor, “Structural stabilization and quenching by dither in nonlinear systems,” IEEE Transactions on Automatic Control , vol. 22, no. 3, pp. 352–361, 1977

    1977

  21. [29]

    Application of a method of averaging to the study of dithers in non-linear systems,

    S. Mossaheb, “Application of a method of averaging to the study of dithers in non-linear systems,” International Journal of Control , vol. 38, no. 3, pp. 557–576, 1983

    1983

  22. [30]

    Dither for smoothing relay feedback systems,

    L. Iannelli, K. H. Johansson, U. Jonsson, and F. Vasca, “Dither for smoothing relay feedback systems,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications , vol. 50, no. 8, pp. 1025–1035, 2003

    2003

  23. [31]

    A framework for control system design subject to average data-rate constraints,

    E. I. Silva, M. S. Derpich, and J. Ostergaard, “A framework for control system design subject to average data-rate constraints,” IEEE Transactions on Automatic Control, vol. 56, no. 8, pp. 1886–1899, 2010

    2010

  24. [32]

    Performance analysis of ran- dom dither quantizers in feedback control systems,

    R. Morita, S.-i. Azuma, and T. Sugie, “Performance analysis of ran- dom dither quantizers in feedback control systems,” IF AC Proceedings V olumes, vol. 44, no. 1, pp. 11296–11301, 2011

    2011

  25. [33]

    Stationary performance evaluation of control systems with random dither quantization,

    K. Kashima and D. Inoue, “Stationary performance evaluation of control systems with random dither quantization,” in European Control Confer- ence, pp. 1625–1630, IEEE, 2014

    2014

  26. [34]

    Effects of quantization and dithering in privacy analysis for a networked control system,

    Y . Kawano and M. Cao, “Effects of quantization and dithering in privacy analysis for a networked control system,” in 60th IEEE Conference on Decision and Control , pp. 2758–2763, IEEE, 2021

    2021

  27. [35]

    Statistical analysis of amplitude-quantized sampled-data systems,

    B. Widrow, “Statistical analysis of amplitude-quantized sampled-data systems,” Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry , vol. 79, no. 6, pp. 555–568, 1961

    1961

  28. [36]

    Characterizing and correcting for the e ffect of sensor noise in the dynamic mode decomposition,

    S. T. Dawson, M. S. Hemati, M. O. Williams, and C. W. Rowley, “Characterizing and correcting for the e ffect of sensor noise in the dynamic mode decomposition,” Experiments in Fluids, vol. 57, pp. 1–19, 2016

    2016

  29. [37]

    Approximating the koopman operator using noisy data: noise-resilient extended dynamic mode decomposition,

    M. Haseli and J. Cort ´es, “Approximating the koopman operator using noisy data: noise-resilient extended dynamic mode decomposition,” in 2019 American Control Conference (ACC) , pp. 5499–5504, IEEE, 2019

    2019

  30. [38]

    Experimental physical parameter estimation of a thyristor driven dc-motor using the hmf-method,

    S. Daniel-Berhe and H. Unbehauen, “Experimental physical parameter estimation of a thyristor driven dc-motor using the hmf-method,”Control Engineering Practice , vol. 6, no. 5, pp. 615–626, 1998

    1998

  31. [39]

    Sen and J

    P. Sen and J. Singer, Large Sample Methods in Statistics (1994): An Introduction with Applications . Boca Raton, Florida: CRC Press, 1993

    1994

  32. [40]

    Asymptotics for minimisers of convex processes,

    N. L. Hjort and D. Pollard, “Asymptotics for minimisers of convex processes,” arXiv preprint arXiv:1107.3806 , 2011. Shahab Ataei is a Ph.D. student in Electrical Engi- neering at The Ohio State University. He received his B.Sc. and M.Sc. degrees in Electrical Engi- neering fr...

    2011 arXiv

Pith tools

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