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

Masashi Wakaiki

REVIEW 1 major objections 4 minor 43 references

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

T0 review · 1 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper establishes a necessary and sufficient LMI condition under which noisy continuous-time input-output data are informative for quadratic stabilization, and proves that any feasible solution yields a controller that stabilizes every

desk verdict The main LMI theorem is sound and genuinely new; referee it, and ask for code/data and a proof of the delegated set equality. read the letter →

arxiv 2602.02992 v2 pith:MAISCJBE submitted 2026-02-03 math.OC cs.SYeess.SY

Masashi Wakaiki This is my paper

classification math.OCcs.SYeess.SY MSC 93D1593B3093C05

keywords data-drivencontrolcontinuous-timesystemsdatainformativityquadraticstabilizationlinearmatrixinequalitiessynthesisoperatorsbehavioralsystemidentification

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 asks when a finite set of noisy continuous-time input-output trajectories is enough to design a stabilizing output-feedback controller for an unknown linear system in autoregressive form, and answers with an exact matrix certificate. It embeds each trajectory into finite-rank synthesis operators, which lets the set of systems consistent with the data be written as a single quadratic matrix inequality. Combining that description with a strict matrix S-lemma and a Lyapunov-style stability condition, it proves that the data are informative for quadratic stabilization if and only if one linear matrix inequality (LMI) is feasible. A feasible solution directly gives a controller that stabilizes all systems consistent with the data. The same operator machinery also characterizes when noise-free data identify the system uniquely.

What carries the argument

The paper's workhorse is the synthesis operator T_ℓ, which maps a test function to a weighted integral of the ℓ-th derivative of the measured trajectory; stacking these operators gives a finite-rank data Hankel operator H and a data-and-noise matrix Π. Lemma 2.1 states that R is consistent with the noisy data exactly when (I_p, R)ᵀΠ(I_p,R) ≥ 0. The strict matrix S-lemma (Lemma 4.2) then converts the requirement 'the stability LMI holds for all such R' into the single LMI (50), with the surjectivity of H ensuring the negative-definiteness and rank conditions the S-lemma needs. The identification result uses the same H: unique recovery of the system from noise-free data is equivalent to H bein

What would settle it

Take a dataset satisfying Assumptions 4.3 and 4.4, solve LMI (50) to get C = DΦ⁻¹, then pick any R consistent with the data (i.e., with (I_p,R)ᵀΠ(I_p,R) ≥ 0) and simulate the closed-loop response from a nonzero initial condition; a bounded or non-decaying trajectory would show the controller does not stabilize all consistent systems. Conversely, if one can exhibit informative data (a stabilizing controller exists for every consistent R) for which LMI (50) is infeasible, the necessity direction fails.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.6: under the assumptions that the data were generated by some system in the noise class (Assumption 4.3) and that the data-embedded operator H is surjective (Assumption 4.4), the data are informative for quadratic stabilization if and only if the LMI (50) is feasible. The matrix N in that LMI is built from products of the synthesis operators and their adjoints, with explicit integral formulas given in Proposition 3.3. Moreover, if the LMI holds with matrices Φ and D, then the controller C = DΦ⁻¹ stabilizes every system in the consistency set Σ_{D,Θ}. The proof hinges on Lemma 2.1, which identifies Σ_{D,Θ} with the set of matrices R satisfying a quadratic inequa

Load-bearing premise

The entire equivalence rests on the data-embedded operator H being surjective, meaning the measured trajectories are rich enough that HH* is invertible; if the data are not that rich, the LMI condition in Theorem 4.6 is not established.

Editorial extensions

If this is right

A practitioner can decide from raw input-output data alone whether stabilization is possible, and obtain a controller from the same LMI, without state or output derivatives and without filtering or intermediate identification.
Because the condition is necessary and sufficient, failure of the LMI certifies that the available data are not informative for quadratic stabilization under the assumed noise bound.
The resulting controller is guaranteed to stabilize every system consistent with the data, so the design is robust to the particular noise realization as long as it lies in the specified class.
In the noise-free limit, the same data operator H determines whether the system coefficients are uniquely identifiable, with an explicit reconstruction formula.
The operator-based embedding applies to the full continuous-time trajectories rather than sampled or filtered proxies, so no pre-processing parameters enter the stability condition.

Reading between the lines

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

If H is not surjective, the paper leaves the informativity question open; in that regime the LMI may be conservative, and one could explore whether a weaker condition, perhaps after adding one more experiment, restores informativeness.
The identification formula R_s = -Y_L H*(HH*)^-1 suggests a derivative-free parameter estimator that needs no basis choice; testing its numerical conditioning against spline-based alternatives would be a natural next step.
The noise class bounds total energy of the embedded noise rather than pointwise size, so the results likely extend to disturbance energies across experiments; it would be interesting to compare with noise models that bound sample paths pointwise.
A direct application of the same S-lemma logic could yield informativity certificates for other objectives, such as H2 or H∞ performance, since the stability certificate is already a quadratic differential form.

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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

Short version: this is a real advance in a narrow area, and the load-bearing theorem holds up. Wakaiki extends the discrete-time behavioral informativity framework to continuous-time AR systems with noisy input-output data by embedding trajectories into synthesis operators, and gets an if-and-only-if LMI for output-feedback stabilization. That is new: earlier continuous-time IO work either used filters and sampling and only got sufficient conditions, or needed state/derivative information. The operator calculus in Section 3 is clean, the proof of Theorem 4.6 is coherent, and the identification side result in Proposition 5.3 is a useful byproduct. The citation pattern is fine; the author builds on [33] and his own [36] rather than hiding them.

Soft spots, in order of size. First, there is no code or data for the LMI example, and the simulation uses Gaussian white noise, which is outside the stated L^2 noise class; the embedded bound is checked after the fact. This does not threaten the theorem, but it weakens the numerical claim. Second, the proof leans on two external results, most visibly the set equality (54), delegated to [31, Theorem 3.4]. The stress-test note fills that gap and it is right: with N22 = -HH* < 0, the top-left block forces the first qL-p columns of any Z to vanish, and the trailing block reproduces Lemma 2.1. Harmless, but the paper should state the argument instead of sending readers elsewhere. Third, Assumption 4.4 (H surjective) is the real content; if H has a nonzero left null vector, then infinitely many R are data-consistent with the same noise and no fixed (Psi,C) can make the quadratic form positive definite for all of them. The paper calls it numerically checkable, which is fair, but it deserves more billing as the core data-richness assumption.

The main algebra checks out, and I could not find a reject-level error. Who is this for: people working in data-driven control, especially continuous-time and behavioral methods. Others can skip the details. Send it to peer review. Referees should ask for code/data and for the proof of (54); the central result deserves referee time.

Referee Report

1 major / 4 minor

Summary. The paper studies data-driven stabilization of continuous-time LTI systems in AR form from noisy input-output trajectories. It embeds the data into synthesis operators (Section 2), characterizes the set of data-consistent systems (Lemma 2.1), derives computable integral formulas for products of synthesis operators and adjoints (Section 3), and proves a necessary and sufficient LMI condition for informativity for quadratic stabilization (Theorem 4.6), under a data-richness assumption on the operator H. A noise-free identification result is also given (Section 5), and a numerical example illustrates the approach (Section 6).

Significance. If Theorem 4.6 is correct, this is the first necessary and sufficient LMI certificate for continuous-time output-feedback stabilization from raw noisy input-output data in the behavioral/AR framework. The synthesis-operator embedding avoids filtering/sampling and yields finite-dimensional matrix conditions. The paper is technically careful: the algebra in Lemmas 2.1, 3.1-3.2, Proposition 3.3, Lemma 4.5, and Theorem 4.6 is coherent, and the use of the strict matrix S-lemma is clearly identified. The noise-free identification result also gives a clean interpretation of the surjectivity assumption on H. The main weaknesses are local: a proof typo in the converse of Theorem 4.6 and the simulation's use of white noise outside the stated L^2/H^L framework.

major comments (1)

[Theorem 4.6, proof of converse] The converse proof defines C:=DPhi and Psi:=Phi^{-1}. With A=[-J_{q(L-1)}Phi; D; 0_{p x qL}], the (1,1) block of the LMI (50) is A^T+A = Phi(-J)^T+(-J)Phi + D^T+D. Substituting D=C Phi^{-1} gives Phi(-J)^T+(-J)Phi + Phi^{-1} C^T + C Phi^{-1}, whereas the corresponding block of M in (51) with Psi^{-1}=Phi is Phi(-J)^T+(-J)Phi + Phi C^T + C Phi. These are not equal. The intended definition is C:=D Phi^{-1}, as already stated in the theorem's controller formula; with that change, A=[-J Phi; C Phi; 0] and A^T+A equals the (1,1)-block of M. Please correct this typo in the proof.

minor comments (4)

[Section 4.2, Eq. (54)] The set equality (54) is delegated to [31, Thm 3.4]. It follows directly from the block structure of N: the (1,1) block of [J_p; Z]^T Pi [J_p; Z] forces the first qL-p columns of any Z in Z_{qL,qL}(N) to vanish because N22=-HH*<0, after which Z=R^T J_p and Lemma 2.1 applies. A two-line derivation would make the main theorem self-contained.
[Section 6, Example] The example applies zero-mean Gaussian white noise with E[v(t)v(s)^T]=delta(t-s)10^{-4}I_2. Such a process is almost surely not an element of L^2([0,tau];R^p), and the resulting output is not in H^L([0,tau];R^p) for L=2. This violates the standing assumptions v_k in L^2 and y_k in H^L from Section 2.1, so the numerical verification of Assumption 4.3 via Proposition 3.3 is not well-defined for the reported noise. Please replace the white noise by a band-limited or colored L^2 approximation, or state explicitly how the white noise is discretized into an L^2 representative.
[Notation] The noise class is introduced as Delta_{tau,Theta} but written as Delta_Theta[0,tau] in Section 6. Please use consistent notation throughout.
[Proposition 3.3] The dimensions of Lambda_ell should be stated explicitly (Lambda_ell in R^{L x n}) to avoid confusion, since tilde f_ell takes values in R^n.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central LMI equivalence is derived from data and external S-lemma results; only minor non-load-bearing self-citations appear.

full rationale

Theorem 4.6 is not circular. The data-consistent set Σ_{D,Θ} is characterized in Lemma 2.1 directly from the defining equations (5) and the noise class, via the identity (I_p R^⊤)^⊤ Π (I_p R^⊤) = Θ − Σ_k (RH_k + Y_{L,k})(RH_k + Y_{L,k})^*. Lemma 4.5 verifies the strict S-lemma hypotheses using only Assumptions 4.3/4.4 and Lemma 2.1. The key set equality (54) is delegated to the external [31, Thm 3.4], not authored by Wakaiki, and the S-lemma itself is [31, Thm 4.10]; these are independent supports. The controller C = DΦ^{-1} in Theorem 4.6 is constructed from the LMI solution, not fitted to a target outcome, and the theorem is an iff statement proven in both directions. The noise-free identification result Proposition 5.3 follows from RH + Y_L = 0 and the surjectivity of H; no target result is assumed. The only self-citations ([36], [37]) concern methodology and a range approximation lemma in Lemma 5.2; they are not load-bearing for the central stabilization theorem, and the cited prior results are consistent with standard operator arguments. The simulation's use of white noise outside the stated L^2 noise class and the reliance on Assumption 4.4 are correctness/reproducibility concerns, not circularity.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The central theorem carries no fitted constants; its inputs are the known noise bound Theta, the known AR order L, and the surjectivity/HH* invertibility assumption. The example adds hand-chosen simulation parameters. The mathematical machinery (synthesis operators, QDF criteria, strict S-lemma) is drawn from prior literature, with two load-bearing external results cited but not reproved.

free parameters (3)

Noise-intensity bound Theta (example) = 10^-6 I_2
Selected by hand in Section 6; the theorem requires Theta known. The actual embedded noise energy computed from the simulated data is <= 7.1e-7 I_2, so the choice is consistent but not derived from first principles.
Trajectory acceptance threshold |theta| <= 0.1 = 0.1 rad
Data selection in Section 6 to keep the linearized model valid. Not used in the theoretical result, but shapes the example dataset.
Data horizon and trajectory count (example) = tau = 0.5, K = 25
Experiment design in Section 6; these are user choices, not fitted to the claim.

assumptions (8)

standard math Strict matrix S-lemma (Lemma 4.2)
Cited from [31, Theorem 4.10] and applied in Theorem 4.6 to convert inclusion of data-consistent systems into existence of a scalar alpha.
standard math Equivalence (54) between R in Sigma and a Z-space of N
Delegated to [31, Theorem 3.4]; load-bearing in the proof of Theorem 4.6 because it re-encodes the data-consistent set as Z_{qL,qL}(N).
standard math Quadratic differential form Lyapunov criterion (Theorem A.1)
Cited from [43]; the basis of Lemma 4.1 and therefore of the definition of quadratic stabilization used in Definition 4.2.
domain assumption The unknown system is exactly a continuous-time AR system of known order L (eq. (2))
The theory assumes no unmodeled dynamics and exact knowledge of L; this is stated in Section 2 but is a real modeling limitation.
domain assumption Noise v_k lies in L^2 and is bounded by known Theta in synthesis-operator norm (Delta_{tau,Theta})
Defines the data-consistent set. The simulation uses Gaussian white noise, which is not literally in the L^2 class; the paper verifies the operator bound a posteriori.
domain assumption Assumption 4.3: some true system R_s and noise sequence in Delta_{tau,Theta} are consistent with the data
Used to prove nonnegativity of the Schur complement in Lemma 4.5. If the chosen Theta is too small, this assumption fails and the theorem is inapplicable.
domain assumption Assumption 4.4: the data-embedded operator H is surjective
Equivalent to HH* invertible; required for N22 < 0 and the strict matrix S-lemma. It is also the identifiability condition in Proposition 5.3.
standard math Density of B-splines in H0^L (Lemma 5.1)
Used in Section 5 to pass from the infinite-dimensional operator H to finite-dimensional B-spline approximations; standard spline theory.

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Pith. "Pith review of Data-driven stabilization of continuous-time systems with noisy input-output data." pith.science (2026). https://pith.science/paper/MAISCJBE

@misc{pith2026260202992,
  author       = {Pith},
  title        = {Pith review of: Data-driven stabilization of continuous-time systems with noisy input-output data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MAISCJBE}},
  note         = {Machine review of arXiv:2602.02992}
}

read the original abstract

We study data-driven stabilization of continuous-time systems in autoregressive form when only noisy input-output data are available. First, we provide an operator-based characterization of the set of systems consistent with the data. Next, combining this characterization with behavioral theory, we establish a necessary and sufficient condition for the noisy data to be informative for quadratic stabilization. This condition is formulated in terms of linear matrix inequalities, whose solutions yield a stabilizing controller. Finally, we characterize data informativity for system identification in the noise-free setting.

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Pith tools

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

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H. J. van Waarde, J. Eising, M. K. Camlibel, and H. L. Trentelman. A behavioral approach to data-driven control with noisy input-output data.IEEE Trans. Automat. Control, 69:813–827, 2023

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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

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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

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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

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Bosso, M

A. Bosso, M. Borghesi, A. Iannelli, B. Yi, and G. Notarstefano. Data-driven stabilization of continuous-time LTI systems from noisy input-output data. arXiv:2511.11417, 2025

arXiv 2025