Direct Data Driven Control Using Noisy Measurements

Gokul S. Sankar; Hamidreza Modares; Ramin Esmzad; Teawon Han

REVIEW 3 major objections 3 minor 23 references

Direct Data Driven Control Using Noisy Measurements

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A single noisy data trajectory can directly tune a linear quadratic regulator.

desk verdict A promising data-driven LQR formulation is undercut by a hidden independence assumption: the proofs treat the gain G as independent of the noise when G is explicitly built from the same noisy data. read the letter →

arxiv 2505.06407 v1 pith:K3264VNA submitted 2025-05-09 eess.SY cs.LGcs.ROcs.SYmath.OC

Ramin Esmzad , Gokul S. Sankar , Teawon Han , Hamidreza Modares This is my paper · ORCID

classification eess.SYcs.LGcs.ROcs.SYmath.OC MSC 93B5293E2090C22

keywords data-drivencontrolnoisystatemeasurementsmean-squarestabilitylinearquadraticregulatorsemidefiniteprogrammingmatrixinequalityinput-outputdatarobust

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 aims to show that a linear quadratic regulator for an unknown linear system can be designed directly from a single trajectory of noisy state measurements, without fitting a model. It introduces an uncertain data-based closed-loop matrix whose mean-square stability is claimed to transfer to the true closed loop (Theorem 1), and then turns stability and optimality into an LMI and an SDP whose solution is a stabilizing feedback gain (Theorems 2 and 3). If the claim holds, this offers a practical path to optimal control in settings where only sensor-corrupted measurements are available and system identification is infeasible.

What carries the argument

The load-bearing device is the parametrization $\begin{bmatrix} K \\ I \end{bmatrix}=D_0G$ with $D_0=\begin{bmatrix} U_0 \\ Y_0 \end{bmatrix}$, which lets any feedback gain $K$ be written as $U_0G$ while enforcing $Y_0G=I$. Substituting this into the noisy data relation produces the uncertain closed-loop matrix $A+BK-A\Upsilon_0G=(Y_1-\Upsilon_1-\Omega_0)G$, and Theorem 1 shows that mean-square stability of this surrogate matrix carries over to the true $A+BK$. The SDP then optimizes over $G$, $M$, and the covariance variables $H,Z,E,S,\Sigma$, enforcing the steady-state covariance equation via an LMI while the remaining constraints encode the parametrization and the linear equations $Y_0F=\Sigma$, $U_0M=I$, $Y_0M=0$.

What would settle it

For a small known system with $n=m=1$, generate one noisy trajectory, solve the SDP (30), and simulate the true closed loop many times with the resulting gain to see whether $\mathbb{E}[x_kx_k^\top]$ converges. Separately, compute the expectation in inequality (16) by Monte Carlo using the actual dependence of $G$ on $\Upsilon_0$; if the true closed loop is not mean-square stable while the SDP is feasible, or the exact expectation violates the inequality used in the proof, the central claim is settled.

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Extended reading notes

Core claim

The central claim is that the unknown closed-loop matrix $A+BK$ can be replaced by the data-only expression $(Y_1-\Upsilon_1-\Omega_0)G$, where $\begin{bmatrix} K \\ I \end{bmatrix}=D_0G$, because mean-square stability of the data-driven matrix $A+BK-A\Upsilon_0G$ implies mean-square stability of the true closed loop. On this basis the paper derives an LMI that certifies MSS directly from noisy data, and an SDP that minimizes the infinite-horizon LQR cost over such gains. Theorem 3 asserts that the optimal gain is $K=U_0F\Sigma^{-1}$, computed purely from one noisy input-output trajectory and the known noise covariances $W$ and $V$, and that this gain stabilizes the true closed-loop system in the mean-square sense.

Load-bearing premise

The load-bearing premise is that the data-derived matrix $G$ can be treated as statistically independent of the measurement noise $\Upsilon_0$ when expected values are taken, even though $G$ is computed from the noisy data matrix $Y_0=X_0+\Upsilon_0$; if that independence is false, the MSS-preservation step and the derived LMI do not follow.

Editorial extensions

If this is right

A controller can be synthesized from a single trajectory of noisy measurements, with no system identification step.
The resulting gain is guaranteed to be mean-square stabilizing for the true closed loop, not only for the data-based surrogate (Theorems 1 and 3).
The synthesis conditions are convex (LMI/SDP), so when a feasible solution exists the gain is obtained by convex optimization.
The noise covariances $W$ and $V$ enter the synthesis directly, avoiding the need to tune ad-hoc robustness parameters.
In the paper's benchmark comparisons, the proposed gain tracks the model-based mean-square LQR gain closely and retains a high success rate even when baselines fail.

Reading between the lines

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

A direct stress test would evaluate the Lyapunov inequality by Monte Carlo with $G$ treated as a function of the noisy data matrix, rather than as a fixed matrix, to see how the MSS certificate behaves as the data length $N$ shrinks.
Because the synthesis needs $W$ and $V$ as inputs, the framework points toward a two-stage pipeline in which the noise covariances are estimated from the same trajectory before the SDP is run; the paper lists covariance estimation as future work.
The same substitution of an uncertain data-only closed-loop matrix for the nominal one could be carried over to other stochastic optimal control problems, such as output-feedback or risk-sensitive LQR, whenever noise statistics are available.

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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

First: the central theorems (1–3) are not supported. The proofs rely on treating the data-dependent matrix G as if it were independent of the measurement noise Υ0 and process noise Ω0 when taking expectations. It isn't: G is defined by Y0G = I with Y0 = X0 + Υ0, so G is a function of Υ0. In Theorem 1, passing from (15) to (16) requires E[Υ0G] = E[Υ0]G and E[GᵀΥ0ᵀAᵀPAΥ0G] = GᵀE[Υ0ᵀAᵀPAΥ0]G, both false in general. The same issue appears in Theorem 2, where Y1 depends on Υ1 and Ω0, so the cross terms with G do not vanish. This is not a minor gap; the MSS-preservation result and the LMI/SDP guarantees rest entirely on it. Without this assumption, a feasible solution to (17) or (30) does not provably give a mean-square stable closed loop.

What the paper does well: the problem is well chosen, and the uncertain closed-loop matrix A+BK−AΥ0G is a clever way to remove the A-dependence. The SDP is a natural convex relaxation, and the simulations on two benchmark systems are fairly extensive and show the method working numerically. The comparison with prior methods is reasonable, and the authors are honest about needing known covariances.

Other soft spots: the final step of Theorem 1 has a small symmetry issue (a skew-symmetric term is dropped), and the derivation of (30f) assumes U0M = I without clear justification until Remark 3. The simulations are not reproducible because no code or data are provided. Also, Theorem 3's optimality is defined relative to the uncertain covariance (29), not the true LQR cost, so even a solved SDP does not give the claimed LQR optimum.

Bottom line: this is a paper for the data-driven control community. The idea is worth thinking about and the SDP may work in practice, but the theoretical guarantee is currently unsupported. I would not trust the stability certificate as stated. If the authors repair the proof—perhaps by conditioning on the data and using a deterministic Lyapunov argument, or by correctly modeling G's randomness—the paper could become solid. As is, I would reject, but it deserves a rigorous referee to document the flaw and give the authors a fair chance at revision.

Referee Report

3 major / 3 minor

Summary. The manuscript considers an unknown discrete-time LTI system with process noise w_k and measurement noise v_k, and assumes access to a single N-step trajectory of inputs and noisy state measurements. It proposes a direct data-driven state-feedback design K = U0G, where G is defined by the data parametrization [K; I] = D0 G with D0 = [U0; Y0]. The authors replace the true closed-loop matrix A+BK by the data-dependent uncertain matrix (Y1 - Υ1 - Ω0)G, claiming that mean-square stability (MSS) of this uncertain matrix implies MSS of the true closed loop (Theorem 1). They then derive an LMI-based stabilization condition (Theorem 2) and an SDP for the linear quadratic regulator (Theorem 3), and report simulations on a quarter-car suspension and a rotary inverted pendulum. The core theoretical claims are that the data-driven design inherits MSS from the uncertain system and that the SDP computes the optimal LQR gain with an MSS guarantee.

Significance. If the three theorems were valid, the paper would make a useful contribution to direct data-driven control under measurement noise: it would provide a convex synthesis procedure with formal MSS and optimality guarantees from a single noisy trajectory, without system identification. The idea of embedding noise statistics into the data-driven closed-loop parametrization is interesting, and the simulation study is extensive. However, the central proofs rely on an unstated and generally false independence assumption between the data-dependent gain G and the noise matrices inside the expectations. This flaw affects Theorems 1-3 and the steady-state covariance formula (29), so the claimed MSS preservation, the LMI condition, and the optimality of the SDP are not established. The paper does not include machine-checked proofs or reproducible code, and the simulations, while suggestive, do not close the theoretical gap.

major comments (3)

[III-A, Theorem 1, Eqs. (15)-(16)] The step from Eq. (15) to Eq. (16) implicitly assumes that G is independent of Υ0 when computing expectations. Specifically, E[Υ0G] = E[Υ0]G and E[G^T Υ0^T A^T P A Υ0 G] = G^T E[Υ0^T A^T P A Υ0]G require G to be treated as a fixed matrix with respect to Υ0. But G is defined by Y0G = I with Y0 = X0 + Υ0 (Eqs. (9b) and (10b)), so G is a function of Υ0. The cross terms that vanish in passing to (16) are generically nonzero, and the quadratic term cannot be factored as written. Consequently, the proof does not establish that MSS of A(I - Υ0G) + BK implies MSS of A + BK.
[III-A, Theorem 2, Eqs. (18)-(19)] The derivation of Eq. (19) from Eq. (18) treats Y1 as a fixed data matrix and G as a fixed gain while taking the expectation over Υ1 and Ω0. However, Y1 is given by Eq. (11), Y1 = AY0 + BU0 - AΥ0 + Υ1 + Ω0, so Y1 is not independent of the noise matrices over which the expectation is taken. The claimed simplification, including the vanishing of cross terms between Y1 and the noises, is therefore unjustified. Moreover, if the expectation is interpreted as conditioning on the observed data, the conditional means of Υ1 and Ω0 are not zero. Thus the LMI (17) is not derived from a valid mean-square stability condition.
[III-B, Eq. (29) and Theorem 3] The steady-state covariance formula (29) is built on the same independence assumption: the expectation is taken over Υ1 and Ω0 while Y1 and G are retained as fixed quantities. In the actual closed-loop system (27), the matrix (Y1 - Υ1 - Ω0) is a fixed realization for the collected data; averaging over the data-generating noise while preserving the measured Y1 is not a well-defined operation and does not yield the covariance of the closed-loop state. Consequently, the SDP (30) does not minimize the LQR cost (22) of the actual closed-loop system, and the optimality and MSS claims in Theorem 3 are not supported by the provided proof.

minor comments (3)

[Table II] For DDNSMLQR (Theorem 3), the Success Count is 150 but Stable Eigenvalues is 121. This discrepancy appears inconsistent with the MSS guarantee claimed in Theorem 3; please clarify whether the eigenvalue check refers to the actual closed-loop system and how it relates to the theoretical guarantee.
[Throughout] The manuscript is formatted as an IEEE TAC paper but the header states 'IEEE T AC, VOL. XX, NO. XX, XXXX 2017' and also references 'IEEE CONTROL SYSTEMS LETTERS'; the template appears inconsistent and should be corrected.
[Theorem 2 and Remark 3] The trace inequality (20) is introduced without derivation, and Remark 3 states that 'Z = E is a trivial solution that satisfies (30f)' without verifying that this choice is compatible with the other constraints; these points should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the controller synthesis is derived from the stated data model and noise statistics, with no fitted parameter renamed as a prediction and no load-bearing self-citation.

full rationale

The paper's derivation chain is self-contained rather than circular. The parametrization (8a), [K; I] = D0 G, is a standard full-row-rank linear algebra fact credited to the independent work [5], and Lemma 1 derives the data equation (9a) directly from the measurement recursion (10)-(11). Theorem 1 is a Lyapunov argument comparing the MSS of A(I - Y0 G) + BK with A + BK; whether the expectation steps are justified is a correctness question about independence of G and Y0, not a circular reduction. Theorem 2 converts the Lyapunov inequality (19) into the LMI (17) via Schur complements and the trace inequality (20), again a direct derivation. Theorem 3 is an SDP reformulation of the covariance expression (29) and the cost (31); the decision variables (Sigma, H, E, F, S, Z, M) are free optimization variables, not quantities fitted to the claimed optimal gain. The only self-citation, [6], is used as a comparison baseline in the simulations and is not invoked to justify any theorem. The paper's genuine weaknesses—treating G as independent of Upsilon0, treating Y1 as fixed while averaging over Upsilon1 and Omega0, and identifying the surrogate covariance (27) with the true LQR cost (22)—are unproven assumptions or overclaims that undermine the proofs, but they do not make the derivation equivalent to its inputs by construction. Under the requirement to flag only specific reductions, no circular step is exhibited.

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

The central claim rests primarily on the stated assumptions (known noise covariances, full rank data, stabilizability/detectability) plus an unstated and generally false independence assumption between the data-derived matrix G and the noise matrices. No physical entities are invented and no free parameters are fitted to data; the SDP decision variables are optimization variables, not fitted parameters.

assumptions (5)

domain assumption The process and measurement noise covariances W and V are known (Assumption 1).
The entire synthesis uses W and V directly in the LMI/SDP; if they are not known, the method cannot be applied as stated, as the paper concedes in Remark 4.
domain assumption The data matrix D0 = [U0; Y0] has full row rank m+n (Assumption 2).
This guarantees existence of G and M satisfying the parametrizations (8a) and (23). With noisy measurements and finite N, this is a persistence-of-excitation condition that may fail.
domain assumption The noise sequences are zero-mean Gaussian and mutually independent across time and of each other and of the initial state.
Used in the expectation computations in Theorems 1 and 2; stated in Section II, but the independence of G from these noises is not stated and is generally false.
ad hoc to paper The data-dependent gain G is treated as independent of the noise matrices in the expectations (implicit in Theorems 1 and 2).
This is an unstated and generally false assumption: G is computed from Y0 = X0 + Υ0, so it is correlated with Υ0. The proofs would need to condition on the data and use posterior noise distributions, which is not done.
standard math (A,B) is stabilizable and (A,√Q) is detectable (Assumptions 3 and 4).
Standard controllability/observability assumptions for LQR.

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Cite this review

Pith. "Pith review of Direct Data Driven Control Using Noisy Measurements." pith.science (2026). https://pith.science/paper/K3264VNA

@misc{pith2026250506407,
  author       = {Pith},
  title        = {Pith review of: Direct Data Driven Control Using Noisy Measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K3264VNA}},
  note         = {Machine review of arXiv:2505.06407}
}

read the original abstract

This paper presents a novel direct data-driven control framework for solving the linear quadratic regulator (LQR) under disturbances and noisy state measurements. The system dynamics are assumed unknown, and the LQR solution is learned using only a single trajectory of noisy input-output data while bypassing system identification. Our approach guarantees mean-square stability (MSS) and optimal performance by leveraging convex optimization techniques that incorporate noise statistics directly into the controller synthesis. First, we establish a theoretical result showing that the MSS of an uncertain data-driven system implies the MSS of the true closed-loop system. Building on this, we develop a robust stability condition using linear matrix inequalities (LMIs) that yields a stabilizing controller gain from noisy measurements. Finally, we formulate a data-driven LQR problem as a semidefinite program (SDP) that computes an optimal gain, minimizing the steady-state covariance. Extensive simulations on benchmark systems -- including a rotary inverted pendulum and an active suspension system -- demonstrate the superior robustness and accuracy of our method compared to existing data-driven LQR approaches. The proposed framework offers a practical and theoretically grounded solution for controller design in noise-corrupted environments where system identification is infeasible.

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Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Data-driven control: Overview and perspec- tives,

W. Tang and P. Daoutidis, “Data-driven control: Overview and perspec- tives,” in 2022 American Control Conference (ACC) , 2022, pp. 1048– 1064

work page 2022

[2] [2]

Robust data-driven output feedback control via bootstrapped multiplicative noise,

B. Gravell, I. Shames, and T. Summers, “Robust data-driven output feedback control via bootstrapped multiplicative noise,” in Learning for Dynamics and Control Conference . PMLR, 2022, pp. 650–662

work page 2022

[3] [3]

Implementation of direct data-driven control on an instrumented bicycle,

A. R ¨onnmark and D. Landr ´e, “Implementation of direct data-driven control on an instrumented bicycle,” 2024

work page 2024

[4] [4]

A practitioner’s guide to noise handling strategies in data-driven predictive control,

A. Sassella, V . Breschi, and S. Formentin, “A practitioner’s guide to noise handling strategies in data-driven predictive control,” IFAC- PapersOnLine, vol. 56, no. 2, pp. 1382–1387, 2023, 22nd IFAC World Congress

work page 2023

[5] [5]

Low-complexity learning of linear quadratic regulators from noisy data,

C. De Persis and P. Tesi, “Low-complexity learning of linear quadratic regulators from noisy data,” Automatica, vol. 128, p. 109548, 2021

2021

[6] [6]

Direct data-driven discounted infinite hori- zon linear quadratic regulator with robustness guarantees,

R. Esmzad and H. Modares, “Direct data-driven discounted infinite hori- zon linear quadratic regulator with robustness guarantees,” Automatica, vol. 175, p. 112197, 2025

work page 2025

[7] [7]

Data-driven output feedback control based on behavioral approach,

Z. Qin and A. Karimi, “Data-driven output feedback control based on behavioral approach,” in 2024 American Control Conference (ACC) , 2024, pp. 3954–3959

work page 2024

[8] [8]

Bridging direct and indirect data-driven control formulations via regularizations and relaxations,

F. D ¨orfler, J. Coulson, and I. Markovsky, “Bridging direct and indirect data-driven control formulations via regularizations and relaxations,” IEEE Transactions on Automatic Control , vol. 68, no. 2, pp. 883–897, 2023

work page 2023