Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Data-driven control of continuous-time systems: A synthesis-operator approach

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read A synthesis-operator identity turns continuous-time trajectory data into exact, derivative-free conditions for identifying and stabilizing unknown linear systems.

desk verdict Genuinely new operator-based framework for derivative-free CT data-driven control; noise-free results solid, noisy-data theorem needs the S-lemma gap closed and a sign typo fixed. read the letter →

arxiv 2511.21041 v2 pith:ERG45WBO submitted 2025-11-26 math.OC cs.SYeess.SY

Masashi Wakaiki This is my paper
classification math.OCcs.SYeess.SY MSC 93B3093D1593C05
keywords data-drivencontrolcontinuous-timesystemssynthesisoperatorsinformativitystabilizationlinearmatrixinequalitiesprocessnoisederivative-freemethods
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 develops a framework for data-driven control of continuous-time linear systems that works directly with measured input-state trajectories, avoiding state derivatives and any sampling or filtering preprocessing. Its central move is to embed the data into 'synthesis operators' on a Sobolev space; integration by parts then converts the differential equation x'=Ax+Bu into the exact operator identity Ξ_d=AΞ+BΥ. On this basis, the paper gives necessary and sufficient conditions for the data to be informative for system identification and for stabilization, and, in the noisy case, an LMI condition for quadratic stabilization that is both necessary and sufficient. A sympathetic reader would care because existing derivative-free methods rely on preprocessing that discards information, so they mostly give sufficient conditions; the operator approach preserves all information in the data and thus characterizes informativity in both directions. If correct, it means continuous-time data-driven control can be decided and synthesized exactly, in parallel with the discrete-time theory.

What carries the argument

The central object is the synthesis operator T associated with a trajectory f, mapping a test function φ in the Sobolev space H^1_0[0,τ] to ∫φ(t)f(t)dt, together with its differentiated counterpart T_dφ=-∫φ'(t)f(t)dt. Integration by parts yields the operator identity Ξ_d=AΞ+BΥ, which encodes the differential equation without state derivatives. Finite-rankness lets range conditions on the operators be replaced by rank conditions on matrices assembled from a piecewise-linear hat-function basis, and explicit integral-kernel formulas convert products of operators into ordinary matrices; the noisy case is handled by the matrix S-lemma applied to a conveniently assembled quadratic form.

What would settle it

Take a scalar unstable system x'=ax+u, generate one noiseless trajectory on [0,τ] with an input that is not persistently exciting (e.g., u≡0), and check the range condition Ran[Ξ;Υ]=R^2; the theory predicts the data are not informative for identification or stabilization. If the range condition is declared full for a trajectory that cannot distinguish a from a+Δa, the framework fails. More directly, add small measurement noise to the state in the paper's batch-reactor example and run the LMI (33); if the resulting K destabilizes the true noiseless plant in closed-loop simulation, the generatio

Watch

Extended reading notes

Core claim

The paper's central claim is that data informativity for continuous-time systems can be characterized exactly in finite-dimensional terms using synthesis operators. For noise-free data satisfying Assumption 2.1, a pair (A,B) is compatible with the data iff Ξ_d=AΞ+BΥ (Lemma 2.2), and the data are informative for system identification iff Ran[Ξ;Υ]=R^{n+m} (Proposition 3.3). The data are informative for stabilization iff RanΞ=R^n and some right inverse Ξ_R^{-1} makes Ξ_dΞ_R^{-1} Hurwitz; the gain is K=ΥΞ_R^{-1} (Theorem 4.1). When the data are corrupted by process noise in Δ_c[0,τ], the data are informative for quadratic stabilization iff the LMI (33) is feasible for some P≻0, L, and α≥0, and t

Load-bearing premise

The whole framework assumes the recorded state and input trajectories are generated exactly by a finite-dimensional linear time-invariant system, possibly with process noise in the prescribed class, with no unmodeled dynamics, time-varying coefficients, or measurement noise on the state; if that generation model fails, the operator identity and all necessary-and-sufficient conclusions are only approximate.

Editorial extensions

If this is right

  • Noise-free data are informative for identification exactly when the combined operator range Ran[Ξ;Υ] fills R^{n+m}; equivalently, for some ℓ, the data matrix built from 2^ℓ-1 hat functions has full row rank.
  • Stabilization from noise-free data reduces to finding a right inverse of Ξ that makes Ξ_dΞ_R^{-1} Hurwitz, with the stabilizing gain read off as K=ΥΞ_R^{-1}.
  • For noisy data with noise class Δ_c[0,τ], feasibility of the LMI (33) is necessary and sufficient for quadratic stabilization, and the same solution yields K=LP^{-1} that works for every compatible (A,B).
  • All conditions are formulated from integrals of the raw trajectories, so no sampling rate, filter, or derivative estimate needs to be chosen.
  • Because the framework treats the data exactly, the resulting informativity conditions are necessary, not just sufficient, matching the strength of the discrete-time informativity theory.

Reading between the lines

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

  • Because the operator-norm noise class Δ_c[0,τ] weights low frequencies more heavily (Proposition 5.2 shows the adjoint norm depends on Fourier coefficients divided by mode number), the framework implicitly tolerates high-frequency noise better than low-frequency noise; one can test whether this matches physical noise models.
  • The finite-dimensional hat-function basis gives a concrete computational route: increase ℓ until the range conditions stabilize, analogous to mesh refinement; a natural extension is to analyze how ℓ trades off against conditioning and LMI feasibility in larger systems.
  • The paper stops at input-state data; extending the operator identity to input-output data (with an observer or a transfer-function representation) would be the natural next step, and the frame-theoretic toolbox seems equipped for it.
  • If exact continuous-time informativity differs from sampled-data informativity, the framework could serve as a benchmark quantifying how much information sampling loses—a comparison that current sufficient-condition methods cannot provide.
Share X Bluesky LinkedIn Reddit HN

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 develops a synthesis-operator framework for data-driven analysis and control of continuous-time LTI systems, avoiding state-derivative measurements. It embeds finite-horizon continuous-time input/state trajectories into finite-rank operators on H_0^1, gives an operator equation (Lemma 2.2) that replaces the differential equation, and characterizes data informativity for system identification and stabilization (Propositions 3.3 and Theorem 4.1). In Section 5, process noise is introduced through a class Δ_c defined by a bound on the synthesis-operator adjoint, and Theorem 5.5 claims a necessary and sufficient LMI condition for informativity for quadratic stabilization, with K=LP^{-1} stabilizing all compatible systems. The numerical example applies the result to a batch-reactor model.

Significance. The noise-free part of the framework is attractive: it is derivative-free, uses the data directly without sampling or filtering, and reduces operator conditions to finite-dimensional matrix conditions via the finite-rank structure. Lemma 2.2, Proposition 3.3, and Theorem 4.1 are carefully argued and appear sound. If the noisy-data result were correct, it would be a useful continuous-time analogue of the discrete-time informativity theory. However, the Section 5 stabilization theorem has serious sign and algebraic errors, and the matrix S-lemma on which its necessity rests is asserted without proof. The contribution therefore cannot be accepted in its present form.

major comments (3)
  1. [Definition 5.1, Eq. (32); Proposition 5.4; Theorem 5.5 proof, Eq. (44)] The Lyapunov inequality in Definition 5.1 has the wrong sign. For a Hurwitz matrix H, the condition should be H P + P H^T ≺ 0 (equivalently H^T P+P H ≺ 0); the stated ≻ 0 condition characterizes eigenvalues in the open right half-plane, not Hurwitz stability. Proposition 5.4 repeats the error: a Hurwitz H does not generally admit P≻0 with H P+P H^T≻0. In the proof of Theorem 5.5, Eq. (44) implies (A+BK)P+P(A+BK)^T ≻ I, which implies A+BK is positive stable, not Hurwitz. The Section 5 definitions, the proof of Proposition 5.4, and the final claim of Theorem 5.5 must be reworked with a consistent negative-definite Lyapunov inequality.
  2. [Lemma 5.6 and proof of Theorem 5.5, (i) ⇒ (ii)] Lemma 5.6 is a nontrivial matrix S-lemma stated without proof or citation, and it carries the entire only-if direction of Theorem 5.5. The hypotheses in (37)-(38) do not imply a Slater/strict-feasibility point for the quadratic matrix inequality; when N22 is singular, no point with [I;Z]^T N [I;Z] ≻ 0 is guaranteed. Standard S-procedure results for matrix variables typically require such a condition or N22 ≺ 0. Without a proof, or a reference with exactly matching hypotheses, the necessity of LMI (33) is unsupported. This is load-bearing because it is the only step that converts the set inclusion Z_{n,n+m}(N) ⊆ Z^+_{n,n+m}(M) into the LMI.
  3. [Theorem 5.5, Eqs. (33), (43)-(44)] There is an algebraic inconsistency between the matrix M defined in (43) and the inequality used in (44) and the LMI (33). Direct computation with M in (43) gives [I;A;B]^T M [I;A;B] = -[(A+BK)^T P + P(A+BK)], not the expression in (44). Moreover, with K=LP^{-1}, the B-dependent terms in the block matrix (33) are of the form KPB and B^T P K^T (through -L B - B^T L^T), whereas the closed-loop Lyapunov expression requires PBK and B^T K^T P. The block dimensions in (33) also appear incompatible with L∈R^{m×n} unless some blocks are transposed. Thus the implication (ii) ⇒ (i) and the claim that K=LP^{-1} makes A+BK Hurwitz do not follow from the printed LMI and proof.
minor comments (4)
  1. [Definition 5.1] Typo: 'quadartic' should be 'quadratic'.
  2. [Section 5.4] The numerical example generates Gaussian white noise. White noise is not an element of L^2([0,τ];R^n), and the resulting state trajectory is not H^1, so Assumption 5.3 is not satisfied. The computation of the approximate norm via Proposition 5.2 also presumes an L^2 noise signal. Please use a smooth or piecewise-polynomial realization in Δ_c, or clarify that the example is an approximation outside the theorem's hypotheses.
  3. [Section 5.4] The reported value c=0.1164 is much larger than the square of the computed operator norm (≈9.6×10^{-2}); the choice of c is not explained. Since a smaller c gives a stronger noise bound, clarify how c=0.1164 was selected.
  4. [Lemma 5.6] If the matrix S-lemma is a known result, a precise citation (e.g., to [26] or a standard reference) should be provided; if it is new, the proof should be included rather than asserted.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the synthesis-operator derivations and LMI characterization are self-contained; the only self-citation is motivational.

full rationale

The central derivation chain is not circular. Lemma 2.2 obtains the operator identity Ξ_d = AΞ + BΥ by integration by parts and the definition of synthesis operators, so the compatible-set representation is derived rather than assumed. Propositions 3.3 and Theorem 4.1 convert operator-range conditions into finite-rank matrix conditions using Douglas' lemma and independently proven approximation lemmas; no fitted parameter is later relabeled as a prediction. In Theorem 5.5, the LMI (33) follows from applying the matrix S-lemma (Lemma 5.6) to the set inclusion Z_{n,n+m}(N) ⊆ Z^+_{n,n+m}(M), where N is constructed from the synthesis operators and the noise bound c. This is a mathematical equivalence step, not a reduction of the theorem to its own input. The only self-citation, [29], appears in the introduction to note that discrete synthesis operators were previously used for discrete-time infinite-dimensional systems; it supplies none of the continuous-time results and is not load-bearing in any proof. The numerical example estimates c from the realized noise before applying the LMI, which is a validation heuristic rather than a derivation that fits the stabilizing gain into the data. One non-circular concern is that Lemma 5.6 is stated without proof or citation, so the necessity direction of Theorem 5.5 depends on an unproved S-lemma; this is a correctness risk, not a circularity. Weighing everything, there is no circular step, only a minor non-load-bearing self-citation and an auxiliary lemma that needs justification.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The framework introduces no new physical entities; synthesis operators are mathematical tools. The main assumptions are the exact LTI generation model and the bounded-noise class, plus unproved background lemmas including the matrix S-lemma. The only free parameters are the noise bound c and the numerical example's solver weights.

free parameters (2)
  • c (noise-intensity bound) = 0.1164 in the numerical example; 0.1 used for the gain computation
    In the theory c is an assumed bound with W W* ⪯ c I_n. In the numerical example c is estimated from the generated noise realization, so the feasibility conclusion depends on this data-derived value.
  • objective weights λ and δ_max = λ=100, δ_max=10^6
    Used only in the numerical example to shape the LMI-solving objective; they do not appear in the theorems and do not affect the central claim.
assumptions (6)
  • domain assumption Assumption 2.1: Data are generated by some A_s,B_s with x'=A_s x+B_s u a.e. on [0,τ].
    All characterizations of Σ_D and informativity rely on exact LTI generation over the observed interval.
  • domain assumption Assumption 5.3: Noisy data are generated with w_s∈Δ_c where W_s W_s* ⪯ c I_n.
    Defines the noise class; the LMI condition is necessary and sufficient only relative to this class.
  • domain assumption State x∈H^1([0,τ];R^n) and input u∈L^2([0,τ];R^m).
    Needed for the integration-by-parts identity in Lemma 2.2; excludes rougher data.
  • standard math Douglas range-inclusion lemma (Lemma 3.1).
    Used in Prop. 3.3 and Thm. 4.1 to pass from range inclusion to factorization and right inverses.
  • standard math Matrix S-lemma (Lemma 5.6).
    Central to Theorem 5.5; asserted without proof or explicit citation, so readers must take it as an unproved background result.
  • standard math Density of piecewise-linear functions in H^1_0[0,τ] and closedness of finite-rank operator ranges.
    Used in Lemma 3.2 and Theorem 4.1 to reduce operator conditions to finite-dimensional matrices.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Data-driven control of continuous-time systems: A synthesis-operator approach." pith.science (2026). https://pith.science/paper/ERG45WBO

@misc{pith2026251121041,
  author       = {Pith},
  title        = {Pith review of: Data-driven control of continuous-time systems: A synthesis-operator approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ERG45WBO}},
  note         = {Machine review of arXiv:2511.21041}
}
read the original abstract

This paper addresses data-driven control of continuous-time systems. We develop a framework based on synthesis operators associated with state and input trajectories. A key advantage of the proposed method is that it does not require the state derivative and uses continuous-time data directly without sampling or filtering. First, systems consistent with the data are represented in terms of synthesis operators, into which the data trajectories are embedded. Next, we characterize data informativity properties for system identification and for stabilization in the noise-free case. Finally, we establish a necessary and sufficient condition for noisy data to be informative for quadratic stabilization. All these informativity characterizations are formulated in terms of finite-dimensional matrices, by leveraging the finite-rank structure of the synthesis operators.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Data-driven stabilization of continuous-time systems with noisy input-output data

    math.OC 2026-02 conditional novelty 6.0 of 10

    Continuous-time noisy input-output data are informative for quadratic stabilization if and only if a data-derived LMI has a feasible solution, whose solution yields a stabilizing controller.

Reference graph

Works this paper leans on

30 extracted references · 2 linked inside Pith · cited by 1 Pith paper

  1. [29]

    M. Wakaiki. Data informativity for stabilization of discrete-time infinite-dimensional systems. arXiv:2511.16008, 2025

    arXiv 2025

  2. [1]

    Alsalti, V

    M. Alsalti, V. G. Lopez, and M. A. M¨ uller. Notes on data-driven output-feedback control of linear MIMO systems.IEEE Trans. Automat. Control, 70:6143–6150, 2025

    2025

  3. [2]

    Axler.Linear Algebra Done Right

    S. Axler.Linear Algebra Done Right. Springer, Cham, 4th edition, 2024

    2024

  4. [3]

    D. S. Bernstein.Scalar, Vector, and Matrix Mathematics: Theory, Facts, and Formulas. Princeton Univ. Press, Princeton, NJ, 2018

    2018

  5. [4]

    Bisoffi, C

    A. Bisoffi, C. De Persis, and P. Tesi. Data-driven control via Petersen’s lemma.Automatica, 145, Art. no. 110537, 14 pp., 2022

    2022

  6. [5]

    Bisoffi, L

    A. Bisoffi, L. Li, C. De Persis, and N. Monshizadeh. Controller synthesis for input-state data with measurement errors.IEEE Control Syst. Lett., 8:1571–1576, 2024

    2024

  7. [6]

    Bosso, M

    A. Bosso, M. Borghesi, A. Iannelli, G. Notarstefano, and A. R. Teel. Data-driven control of continuous-time LTI systems via non-minimal realizations. arXiv:2505.22505, 2025

    arXiv 2025

  8. [7]

    Bosso, M

    A. Bosso, M. Borghesi, A. Iannelli, G. Notarstefano, and A. R. Teel. Derivative-free data-driven control of continuous-time linear time-invariant systems.Eur. J. Control, Art. no. 105965, 8 pp., 2025

    2025

Show all 30 references
  1. [8]

    Christensen.An Introduction to Frames and Riesz Bases

    O. Christensen.An Introduction to Frames and Riesz Bases. Birkh¨ auser, Cham, 2nd edition, 2016

    2016

  2. [9]

    De Persis, R

    C. De Persis, R. Postoyan, and P. Tesi. Event-triggered control from data.IEEE Trans. Automat. Control, 69:3780–3795, 2024

    2024

  3. [10]

    De Persis and P

    C. De Persis and P. Tesi. Formulas for data-driven control: Stabilization, optimality, and robustness.IEEE Trans. Automat. Control, 65:909–924, 2020

    2020

  4. [11]

    R. G. Douglas. On majorization, factorization, and range inclusion of operators on Hilbert space.Proc. Amer. Math. Soc., 17:413–415, 1966

    1966

  5. [12]

    Eising and J

    J. Eising and J. Cort´ es. When sampling works in data-driven control: Informativity for stabilization in continuous time.IEEE Trans. Automat. Control, 70:565–572, 2025

    2025

  6. [13]

    H. Gao, A. Bosso, L. Wang, D. Saussi´ e, and B. Yi. Input-output data-driven stabilization of continuous-time linear MIMO systems. arXiv:2511.06524, 2025

    2025

  7. [14]

    Kaminaga and H

    T. Kaminaga and H. Sasahara. Data informativity for quadratic stabilization under data perturbation. arXiv:2505.01641, 2025

    2025 arXiv

  8. [15]

    A. Koch, J. Berberich, and F. Allg¨ ower. Provably robust verification of dissipativity properties from data.IEEE Trans. Automat. Control, 67:4248–4255, 2022

    2022

  9. [16]

    M. G. Larson and F. Bengzon.The finite element method: Theory, implementation, and applications. Springer, Berlin, 2013

    2013

  10. [17]

    L¨ ofberg

    J. L¨ ofberg. YALMIP: A toolbox for modeling and optimization in MATLAB. InProc. IEEE ICRA 2004, pages 284–289, 2004

    2004

  11. [18]

    Version 11.0.29, 2025

    MOSEK ApS.MOSEK API for MATLAB manual. Version 11.0.29, 2025

    2025

  12. [19]

    Y. Ohta. Data informativity of continuous-time systems by sampling using linear functionals. InProc. MTNS 2024, 6 pp., 2024

    2024

  13. [20]

    Ohta and P

    Y. Ohta and P. Rapisarda. A sampling linear functional framework for data-driven analysis and control of continuous-time systems. InProc. 63rd IEEE CDC, pages 357–362, 2024

    2024

  14. [21]

    Possieri

    C. Possieri. A derivative free framework for data-driven stabilization of continuous-time linear systems.IEEE Control Syst. Lett., 9:1219–1224, 2025

    2025

  15. [22]

    Rapisarda, H

    P. Rapisarda, H. J. van Waarde, and M. K. C ¸ amlibel. Orthogonal polynomial bases for data-driven analysis and control of continuous-time systems.IEEE Trans. Automat. Control, 69:4307–4319, 2024

    2024

  16. [23]

    H. H. Rosenbrock.Computer-Aided Control System Design. Academic Press, London, 1974

    1974

  17. [24]

    Song and J

    C. Song and J. Liu. All data-driven LQR algorithms require at least as much interval data as system identification.IEEE Control Syst. Lett., 9:1778–1783, 2025

    2025

  18. [25]

    Tucsnak and G

    M. Tucsnak and G. Weiss.Observation and Control of Operator Semigroups. Birkh¨ auser, Basel, 2009

    2009

  19. [26]

    H. J. van Waarde, M. K. Camlibel, J. Eising, and H. L. Trentelman. Quadratic matrix inequalities with applications to data-based control.SIAM J. Control Optim., 61:2251–2281, 2023

    2023

  20. [27]

    H. J. van Waarde, J. Eising, M. K. Camlibel, and H. L. Trentelman. The informativity approach: To data-driven analysis and control.IEEE Control Syst. Mag., 43:32–66, 2023

    2023

  21. [28]

    H. J. van Waarde, J. Eising, H. L. Trentelman, and M. K. Camlibel. Data informativity: A new perspective on data-driven analysis and control.IEEE Trans. Automat. Control, 65:4753–4768, 2020

    2020

  22. [30]

    J. C. Willems, P. Rapisarda, I. Markovsky, and B. L. De Moor. A note on persistency of excitation.Syst. Control Lett., 54:325–329, 2005. Graduate School of System Informatics, Kobe University, Nada, Kobe, Hyogo 657-8501, Japan Email address:wakaiki@ruby.kobe-u.ac.jp

    2005

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.