{"id":"26ae6701-9248-4694-8d7b-83adf7caa551","arxiv_id":"2505.06407","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors derive LMI and SDP conditions for data-driven LQR from noisy state measurements, but the proofs rely on unstated independence assumptions and the optimality claim is not tied to the true system.","lead":"This paper proposes a way to design a feedback controller directly from one noisy data trajectory of an unknown linear system, skipping system identification. It promises a stability guarantee and near-optimal performance, with simulations on a suspension system and an inverted pendulum.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central stability and optimality claims rely on an unstated independence of the data-dependent gain G from the noise matrices when taking expectations; because G is built from Y0=X0+Υ0, this independence is generically false, so Theorems 1–3 do not follow as stated.","rationale":"The reader's weakest-assumption identification is correct and I agree with the rejection. The independence of G from the noise matrices is not a minor technicality: it is the only bridge between the data-driven uncertain matrix (13) and the true closed-loop matrix A+BK. Without it, feasibility of SDP (17) or (30) says nothing about MSS of (3). Theorem 3's optimality claim is also a casualty, because the covariance expression (29) is derived under the same independence, so the objective in (30h) is not a valid expression for J(K) unless G and M are exogenous. This is an internal inconsistency with the paper's own data model: equations (10a)-(11) make Ω0 enter X1 and Y1, while Y0 enters D0 and hence G, so G and Ω0 are dependent under the stated Gaussian assumptions. I also note a further gap in Theorem 3's proof: the 'equivalence' of H=FΣ^{-1}F^T and E=FS^{-1}F^T with the LMIs (30c)-(30d) is only a relaxation, so even with independence the optimality step is not established. But the independence issue is prior, since the MSS guarantee comes from Theorems 1-2. Hence the verdict should remain REJECT.","tokens_in":13886,"tokens_out":19248,"duration_ms":192243,"concrete_test":"Simulate the paper's model (10)-(11) for a scalar system with A=1.1, B=1, N=2, W=V=0.01, a fixed P>0 and candidate K. For each of S=10^5 draws of x0, ω0, ω1, υ0, υ1, compute Y0, Y1, G=D0^{-1}[K;1] (so Y0G=1), and D=G^T(Y1−Υ1−Ω0)^T P(Y1−Υ1−Ω0)G − G^T Y1^T P Y1 G − G^T Tr(P(W+V))G. If the sample mean of D is not zero, the cross terms dropped in (19) are non-vanishing and the independence step in Theorem 2 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central guarantee is that MSS of the data-driven uncertain matrix (13) transfers to the true closed loop (Theorem 1), and that the LMIs/SDP in Theorems 2 and 3 can be built on this transfer. The proof steps that take expectations over the latent noise matrices treat the gain G as a fixed, non-random matrix. In Theorem 1, passing from (15) to (16) requires 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, i.e. G independent of Υ0; but G is defined by (8a), Y0G=I, with Y0=X0+Υ0, so G is a function of Υ0. In Theorem 2, the step from (18) to (19) requires E[G^T Ω0^T P Ω0 G]=G^T Tr(PW)G and vanishing cross terms such as E[G^T Y1^T P Ω0 G]. The data model (10)-(11) makes Y0 (hence G) depend on past entries of Ω0 through X1=AX0+BU0+Ω0, so these equalities fail in general. The covariance formula (29) and the SDP objective (30h) rest on the same assumption. If one instead conditions on the observed data, Υ1 and Ω0 are fixed unknown realizations and the marginal Gaussian expectations used in (19) are unjustified. Thus Theorems 1–3 do not establish MSS preservation or optimality for the actual closed-loop system.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":14321,"tokens_out":7836,"duration_ms":76365,"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":[{"comment":"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.","section":"III-A, Theorem 1, Eqs. (15)-(16)"},{"comment":"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.","section":"III-A, Theorem 2, Eqs. (18)-(19)"},{"comment":"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.","section":"III-B, Eq. (29) and Theorem 3"}],"minor_comments":[{"comment":"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.","section":"Table II"},{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Theorem 2 and Remark 3"}],"recommendation":"reject","confidential_remarks":"The central technical flaw is not a local fix: the proofs of Theorems 1-3 all hinge on treating the data-dependent gain G as independent of the measurement noise in the expectations, which is false by construction. Repairing this would require a substantially different analysis, possibly with high-probability or deterministic data-informativity arguments, and would change the scope of the paper. I therefore recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nOther 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.\n\nBottom 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.","headline":"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.","tokens_in":14783,"tokens_out":6229,"would_cite":false,"duration_ms":55229,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B52","93E20","90C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single noisy data trajectory can directly tune a linear quadratic regulator.","keywords":["data-driven control","noisy state measurements","mean-square stability","linear quadratic regulator","semidefinite programming","linear matrix inequality","input-output data","robust control"],"falsifier":"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.","tokens_in":13749,"feed_emoji":"🎛️","tokens_out":10075,"duration_ms":89183,"temperature":0.7,"pith_summary":"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.","feed_headline":"One noisy data run can tune an LQR controller","feed_subtitle":"New method claims mean-square stability and near-optimal gains without system identification.","key_machinery":"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 $Y_0F=\\Sigma$, $U_0M=I$, $Y_0M=0$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the data parametrization [K; I] = D0 G and the low-complexity LQR baseline that this paper extends to noisy measurements.","marker":"[5]"},{"why":"Is the discounted data-driven LQR baseline used for comparison and motivates the Bellman-style cost structure behind the SDP.","marker":"[6]"},{"why":"Represents the earlier noisy-data approach that bounds the lumped uncertainty, against which the paper positions its reformulation.","marker":"[15]"},{"why":"Is the prior robust LQG-from-data method requiring noise-free offline trajectories, the gap this paper addresses.","marker":"[14]"},{"why":"Provides the mean-square stability definition and Lyapunov characterization on which Theorems 1 and 2 rely.","marker":"[18]"}],"fun_headline_variants":["Noisy data, single run: LQR solved directly","Bypass identification: LQR from one noisy run","Direct LQR from noisy data with stability guarantee","One noisy trajectory yields stabilizing LQR gain","Data-driven LQR without system ID, despite noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Noisy data, single run: LQR solved directly","Bypass identification: LQR from one noisy run","Direct LQR from noisy data with stability guarantee","One noisy trajectory yields stabilizing LQR gain","Data-driven LQR without system ID, despite noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000241,"raw_usage":{"total_tokens":1505,"prompt_tokens":913,"completion_tokens":592,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":529,"completion_tokens_details":{"reasoning_tokens":517}},"tokens_in":529,"tokens_out":592,"duration_ms":5322,"temperature":1.0,"reasoning_tokens":517,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:45:07.676031+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Direct data-driven discounted infinite hori- zon linear quadratic regulator with robustness guarantees,","cited_arxiv_id":null,"evidence_quote":"Is the discounted data-driven LQR baseline used for comparison and motivates the Bellman-style cost structure behind the SDP."},{"cited_title":"Learning robust data- based lqg controllers from noisy data,","cited_arxiv_id":null,"evidence_quote":"Is the prior robust LQG-from-data method requiring noise-free offline trajectories, the gap this paper addresses."},{"cited_title":"Mean square stability conditions for discrete stochastic bilinear systems,","cited_arxiv_id":null,"evidence_quote":"Provides the mean-square stability definition and Lyapunov characterization on which Theorems 1 and 2 rely."}],"review_version":1}