{"id":"37d51f08-0c90-42a3-9a11-767ec5d98dbf","arxiv_id":"2505.22505","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A single-experiment, derivative-free data-driven method for stabilization and output regulation of MIMO continuous-time LTI systems using non-minimal realizations.","lead":"This paper designs output-feedback controllers for unknown continuous-time linear systems using only input-output data from a single experiment. It builds filters that reconstruct a higher-dimensional state and solves linear matrix inequalities to compute stabilizing or regulating control gains.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central stabilization proof is sound conditional on a canonical realization, but the only general construction requires uniform observability indices, restricting the MIMO claim to a non-generic class.","rationale":"The reader's weakest-assumption analysis correctly identifies Assumption 3 as the structural bottleneck. My independent check of Theorem 1 found no flaw in the LMI argument: the rank condition (38) supplies the right inverse Zdagger, the unknown D is eliminated via XQ=0, and the Lyapunov inequality (46) indeed makes F+LH+GK Hurwitz. Lemma 4 is also sound given that the modes of A-PiLC are contained in the modes of F0. The problem is not internal consistency but scope: the paper's construction of the canonical non-minimal realization, and hence the entire derivative-free output-feedback framework, is proven only when all observability indices coincide. Since generic MIMO systems have distinct observability indices, the central contribution does not cover the general MIMO case advertised in the title and introduction. The omitted proof of Theorem 2 is a lesser concern because the internal-model reduction is standard once the stabilizability of the augmented pair is established via Corollary 2. The full-rank condition in Algorithm 3 is also assumed rather than derived from an explicit excitation condition, reinforcing the conditional nature of the practical procedure. These are addressable but real gaps, so the CONDITIONAL verdict is appropriate.","tokens_in":29855,"tokens_out":18213,"duration_ms":243941,"concrete_test":"Implement the proposed construction for a third-order two-output plant with observability indices (1,2), e.g. A in observer canonical form with nu_1=1, nu_2=2 and any controllable B. Set nu=2 in Theorem 3 and attempt to solve equation (13) with F, G, L from (78). If no Pi, H solve the decoupled linear equations (113)-(116), this confirms that the uniform-index construction cannot be patched by a mere parameter choice and that a genuinely different block structure is needed for nonuniform observability indices.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1 and the LMI argument are internally consistent: given F, G, L such that (12) is a canonical non-minimal realization, the data equation (37) and the right-inverse step in (44)-(45) go through, and the numerical examples are constructive evidence. The load-bearing gap is the construction of the realization. The paper provides a construction only under Assumption 3 (nu_1=...=nu_p=nu). This is not a minor technical condition: for generic MIMO pairs (C,A), observability indices are distinct, so the class is non-generic. Theorem 3 solves (116) by decoupling into identical nu-dimensional blocks; for nonuniform nu_i the algebraic equations do not decouple in that way, and the paper offers no alternative tuning. Thus the headline claim of data-driven output-feedback control for MIMO LTI systems is only established for a restricted class. Moreover, Algorithm 3's observability-index estimator inherits the same restriction: Theorem 5 assumes full rank of B at nuhat=nu rather than proving it from a persistency-of-excitation condition, so even within Assumption 3 the index estimation step is not certified. This does not invalidate Theorem 1, but it moves the practical scope from 'MIMO' to 'uniform-observability-index MIMO'.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a data-driven output-feedback control framework for continuous-time LTI systems using non-minimal realizations whose state is obtained by filtering input and output signals, avoiding state and derivative measurements. For stabilization, the paper derives an LMI from sampled filtered data and proves that any feasible solution yields a stabilizing dynamic controller (Theorem 1). The framework is extended to output regulation by adding an internal model based on known exosystem frequencies (Theorem 2). A construction of canonical non-minimal realizations is given for systems with uniform observability index (Theorem 3), together with controllability of the resulting realization (Theorem 4) and an algorithm for estimating the observability index from data (Algorithm 3, Theorem 5). Numerical examples on a batch reactor and a surface vessel illustrate the approach; code is provided.","tokens_in":30142,"tokens_out":20888,"duration_ms":226415,"significance":"If the identified gaps are closed, this would be a valuable contribution to data-driven control of continuous-time systems: it provides a derivative-free, single-trajectory, output-feedback method for MIMO LTI systems, with an extension to output regulation and a filter-based implicit observer. Theorem 1's proof is detailed and internally consistent, and the LMI derivation is non-circular. The restriction to uniform observability index and the noise-free assumption are explicitly acknowledged, and the numerical implementation is reproducible via the provided repository. The main weaknesses are proof gaps in the output-regulation theorem and in the index-estimation certification.","major_comments":[{"comment":"The output-regulation theorem is stated without proof (\"Its proof is omitted as it is based on Theorem 1 and the arguments above\"), and the essential extension of Lemma 4 to the exosystem-augmented dynamics is also omitted (\"The statement of such result and its proof are omitted for brevity\"). These omissions are load-bearing: the data equation behind LMI (53), the rank condition (67), and the conclusion that the matrix in (68) is Hurwitz are not verifiable from the text. Please provide a complete proof in an appendix, including the explicit construction of D in col(w,ε)=Dχ, the derivation of the data equation from (62), and the verification of closed-loop regulation property (11) via (61).","section":"Section V, Theorem 2"},{"comment":"Theorem 1 is stated under Assumption 1 alone, but its initialization requires a canonical non-minimal realization, and the only general construction (Theorem 3) is proved under Assumption 3 (ν1=...=νp=ν). For generic MIMO systems the observability indices are as evenly balanced as possible, so uniform indices are possible only when p divides n; when p does not divide n, Assumption 3 cannot hold. Equations (116) decouple into identical ν-dimensional blocks precisely because of uniformity, and no alternative tuning is given for nonuniform indices. Thus the paper's MIMO stabilization and regulation claims are established only for the uniform-index class. Please state this restriction in Section II-A and in the abstract/contribution list, not only in Section VI and the conclusion.","section":"Section VI-C and Section II-A"},{"comment":"The index-estimation procedure is not fully certified. Theorem 5 proves rank deficiency for ν̂≥ν+1, but the full-rank condition at ν̂=ν is assumed, not proved: Algorithm 3 contains the comment \"Assumes B full rank if ν̂=ν\", and the text preceding Theorem 5 says \"Suppose that, for ν̂=ν, B has full rank\". No persistency-of-excitation or sampling condition is given under which this full-rank condition holds. Since this full-rank condition is necessary for the subsequent LMI feasibility condition (38), the claim that the observability index \"can be directly estimated from the given input-output trajectory\" needs either a proof under explicit excitation assumptions or a clear downgrade to a heuristic/assumption.","section":"Section VI-D, Algorithm 3 and Theorem 5"}],"minor_comments":[{"comment":"The text introduces observability indices under the condition rank C=p, but Assumption 1 alone does not guarantee this when p>n. This implicit standing assumption should be stated explicitly in Section II.","section":"Section VI-C"},{"comment":"In part 1, the existence of M from the rank condition (38) is asserted by reference to [4] and [19]; a one-line argument that full row rank of [X;Z;U] makes the representation (40) possible would make the proof self-contained.","section":"Section IV, proof of Theorem 1"},{"comment":"The display (136) is typeset with blank entries and no explicit block dimensions; an explicit 4x4 block matrix with zero blocks and dimensions would substantially improve readability.","section":"Section VI-D, proof of Theorem 5"},{"comment":"The paper relies on [25] for Lemma 1 and the filter structure; although Lemma 1 is proved in the text, a brief remark on how the present results go beyond [25] (from SISO/state-feedback to MIMO and output regulation under the uniform-index assumption) would help the reader.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the journal's scope and the core LMI argument is sound. The main risks are the omitted proof of Theorem 2 and the unproven full-rank condition in Algorithm 3. The uniform-index restriction is a genuine scope limitation but is explicitly acknowledged; I would not reject on that basis alone. The dependence on the authors' own preprint [25] for the central filter structure should be monitored for overlap with the prior work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a look if you work on direct data-driven control in continuous time. The paper extends the SISO/state-feedback framework from the authors' earlier work [25] to MIMO output-feedback stabilization, adds output regulation via internal models, and proposes an observability index estimator. The core trick is still the filtered non-minimal realization plus an LMI on sampled data, but the MIMO generalization is not just notation pushing: Theorem 1 is proved in detail, and Theorem 3 in the appendix gives a real construction of the realization. The numerical examples come with code and match the theory. No circularity — the self-citation to [25] is legitimate because the SISO result is the foundation they are building on.\n\nNow the soft spots, in order of softness. The uniform observability index assumption (Assumption 3) is structural, not technical. For generic MIMO output matrices, observability indices are distinct, so the headline claim of MIMO output-feedback control only covers a restricted, non-generic class. The paper does state this as future work, but it should be surfaced in the title or abstract — otherwise a reader could overestimate the scope. Second, Theorem 2, the output regulation result, is stated without proof ('omitted for brevity'). The argument is plausibly a direct analog of Theorem 1, but as submitted the key theorem of Section V is unverified. Third, Algorithm 3 only proves the negative direction (rank loss for nuhat > nu); it assumes rather than proves the full-rank condition at the true index. No persistency-of-excitation style proof is given, so the estimator is not fully certified. That is the weakest link in the claimed contribution.\n\nNone of this sinks the paper. The central stabilization logic is sound given a canonical realization, and the limitation is clearly acknowledged in the text. It is a solid contribution for the class it actually handles, and the readers who will get the most value are people working on derivative-free continuous-time data-driven control and output regulation.\n\nMy recommendation: send it to peer review. A serious referee should ask for a proof or a complete reference for Theorem 2, a sharper treatment of the full-rank condition in Algorithm 3, and an honest reframing of the MIMO contribution so the uniform-index restriction is front and center. With those changes it could be a solid journal paper.","headline":"Solid extension of the authors' prior SISO work to MIMO output-feedback stabilization and regulation, but the MIMO claim narrows substantially under the uniform observability index assumption.","tokens_in":30664,"tokens_out":2377,"would_cite":true,"duration_ms":29508,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B52","93B30","93C05","93C35","93D15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single noise-free trajectory of a continuous-time LTI plant suffices to design a stabilizing output-feedback controller, and the same construction solves output regulation when all outputs share one observability index.","keywords":["data-driven control","continuous-time LTI systems","output feedback","non-minimal realizations","linear matrix inequalities","output regulation","observability indices","MIMO systems"],"falsifier":"Choose a two-input, two-output continuous-time LTI plant with uniform observability index $\\nu=2$, collect one noise-free trajectory, and run Algorithm 1 with $N$ samples satisfying the rank condition (38) and Algorithm 3 on the same data. If the computed gain $K$ ever leaves $F+LH+GK$ with an eigenvalue in the closed right half-plane, or if the index estimate differs from 2, then the paper's central claim is false.","tokens_in":29669,"feed_emoji":"⚙️","tokens_out":10629,"duration_ms":99338,"temperature":0.7,"pith_summary":"This paper claims that output-feedback controllers for unknown continuous-time linear time-invariant systems can be designed directly from one input-output experiment, without measuring states or derivatives. The key move is to use a canonical non-minimal realization: the user chooses stable filter dynamics with the unknown plant confined to the output equation, then reconstructs its state by filtering the measured signals. From sampled batches of filtered data, a linear matrix inequality computes a gain that provably makes the closed loop exponentially stable, and the same construction, augmented with an internal model driven by the known exosystem frequencies, solves output regulation. The method works for MIMO plants whose observability indices are all equal, and the paper provides a data-based test to estimate that common index. If true, the result removes derivative measurements—the main obstacle to continuous-time data-driven output feedback—from both stabilization and regulation problems.","feed_headline":"One trajectory yields stabilizing gains with no state or derivatives","feed_subtitle":"The controller and its regulator variant come from one sampled batch, with no derivatives and no plant matrices.","key_machinery":"The central object is the canonical non-minimal realization, a lifted system $\\dot\\zeta=(F+LH)\\zeta+Gu$, $y=H\\zeta$ in which $F,G,L$ are user-chosen and $F$ is Hurwitz, while the unknown plant parameters are absorbed into the output map $H$, which must satisfy the matrix equations (13) with some $\\Pi$. The non-minimal state is reconstructed by the filter $\\dot{\\hat\\zeta}=F\\hat\\zeta+Gu+Ly$; because $F$ is Hurwitz, Lemma 3 makes the error dynamics $A-\\Pi LC$ stable, and Lemma 4 replaces the unmeasurable error with a user-simulated auxiliary signal $\\chi$ obeying $\\dot\\chi=F_0\\chi$. This yields sampled batches $X,Z,\\dot Z,U$ linked by the differential relation (37), so the LMI (29) is a pure data condition: its feasibility is equivalent to the full-rank condition (38), and its solution gives the gain $K=UQP^{-1}$. For MIMO plants with uniform observability index, Theorem 3 supplies explicit filter gains (78) built from a controllable pair $(\\Lambda,\\ell)$, and Algorithm 3 estimates $\\nu$ by the rank loss of a data batch (86).","core_discovery":"The paper's central claim is that stabilization and output regulation in continuous time can be cast as a rank-conditioned linear matrix inequality built from filtered data. Given a canonical non-minimal realization—user-chosen Hurwitz filter dynamics $F,G,L$ with unknown output map $H$ satisfying $\\Pi(F+LH)=A\\Pi$, $\\Pi G=B$, $H=C\\Pi$—the paper proves (Theorem 1) that if the stacked data matrix of the sampled batches $X,Z,U$ has full rank, then any solution $P,Q$ of the LMI (29) yields $K=UQP^{-1}$ such that $F+LH+GK$ is Hurwitz, so the observer-like filter (31) globally exponentially stabilizes the plant. For regulation, an internal model $\\dot\\eta=\\Phi\\eta+\\Gamma e$ is appended and the same LMI construction is applied to the augmented system; Theorem 2 states that the resulting controller (55) steers the regulated output $e$ to zero under the non-resonance condition. A data-driven algorithm (Algorithm 3) recovers the common observability index from the same trajectory, so the realization can be tuned from data. In short: a stabilizing output-feedback controller for a continuous-time MIMO LTI system can be computed directly from a single noise-free input-output trajectory without state or derivative measurements.","pith_inferences":["The paper's restriction to a uniform observability index is likely removable: following the discrete-time MIMO literature it cites, a filter bank with per-output dimensions $\\nu_i$ and a similar rank test should extend the construction to generic MIMO systems, though the present proofs do not cover that case.","Because the LMI depends only on filtered data batches, replacing the exact batch equations with robust matrix relaxations for bounded noise is a natural route to measurement noise; the filter and internal-model structure would remain unchanged.","The auxiliary-dynamics trick of Lemma 4 turns a differential equation with an unknown perturbation into a purely data-driven equality, so the same device could reduce other continuous-time data-driven problems, such as optimal or constrained control, to LMIs."],"forward_implications":["Any controllable and observable continuous-time LTI plant with a uniform observability index can be stabilized from one noise-free trajectory: the LMI (29) is feasible whenever the sampled batch satisfies the rank condition (38), and Theorem 1 guarantees the resulting closed loop is Hurwitz.","The deployed controller has an observer-plus-gain form (31), so its online implementation uses only the same filters and the computed gain; no model matrices or derivative measurements are needed after deployment.","For output regulation, given the exosystem frequencies and the non-resonance condition, the same LMI applied to the augmented system produces the internal-model regulator (55), and Theorem 2 guarantees the regulated output converges to zero.","The observability index can be read off the data: if the batch is full rank at $\\hat\\nu=\\nu$, then increasing the filter dimension makes the batch lose rank (Theorem 5), so Algorithm 3 returns the correct uniform index.","The construction includes state-feedback ($\\nu=1$) and SISO output-feedback ($\\nu=n$) as special cases, so it unifies two previously separate derivative-free designs."],"supporting_citations":[{"why":"The companion derivative-free design whose filter-based non-minimal realization this paper extends to MIMO and output regulation.","marker":"[25]"},{"why":"The sampled-batch LMI construction for continuous-time data-driven control, whose inequalities (29) and (53) directly adapt.","marker":"[19]"},{"why":"The data-driven formula and rank-condition proof technique used in Theorem 1.","marker":"[4]"},{"why":"The internal-model output regulation machinery, including the post-processing internal model, the non-resonance condition, and the regulator structure, underpinning Algorithm 2.","marker":"[21]"},{"why":"The source for observability indices, observer canonical forms, the PBH test, and polynomial matrix descriptions used in Theorems 3 through 5.","marker":"[34]"},{"why":"The adaptive-observer filter structure that motivates the SISO tuning (73) and its MIMO generalization.","marker":"[27]"},{"why":"Discrete-time MIMO output-feedback controller synthesis where observability indices play the same role, indicating the intended generalization.","marker":"[12]"}],"fun_headline_variants":["One sampled batch stabilizes LTI systems without state or derivatives","Filtered data alone yields output-feedback controllers for unknown LTI","Single trajectory, LMI-based control: no state, no derivatives","Data-driven regulators from one input-output run, no plant matrices","Output feedback from filtered data: stabilize LTI without derivatives"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on every output having the same observability index $\\nu$; if the indices differ, the tuning rules and the index-estimation algorithm are not known to work, and the paper leaves that case open.","fun_headline_variants_meta":{"raw":{"variants":["One sampled batch stabilizes LTI systems without state or derivatives","Filtered data alone yields output-feedback controllers for unknown LTI","Single trajectory, LMI-based control: no state, no derivatives","Data-driven regulators from one input-output run, no plant matrices","Output feedback from filtered data: stabilize LTI without derivatives"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000186,"raw_usage":{"total_tokens":1327,"prompt_tokens":946,"completion_tokens":381,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":294}},"tokens_in":562,"tokens_out":381,"duration_ms":5077,"temperature":1.0,"reasoning_tokens":294,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:04:58.180886+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose a two-input, two-output continuous-time LTI plant with uniform observability index $\\nu=2$, collect one noise-free trajectory, and run Algorithm 1 with $N$ samples satisfying the rank condition (38) and Algorithm 3 on the same data. If the computed gain $K$ ever leaves $F+LH+GK$ with an eigenvalue in the closed right half-plane, or if the index estimate differs from 2, then the paper's central claim is false.","supporting_citations":[{"cited_title":"Derivative-Free Data-Driven Control of Continuous-Time Linear Time-Invariant Systems","cited_arxiv_id":"2410.24167","evidence_quote":"The companion derivative-free design whose filter-based non-minimal realization this paper extends to MIMO and output regulation."},{"cited_title":"Data-driven harmonic output regulation of a class of nonlinear systems,","cited_arxiv_id":null,"evidence_quote":"The sampled-batch LMI construction for continuous-time data-driven control, whose inequalities (29) and (53) directly adapt."},{"cited_title":"Isidori, Lectures in Feedback Design for Multivariable Systems","cited_arxiv_id":null,"evidence_quote":"The internal-model output regulation machinery, including the post-processing internal model, the non-resonance condition, and the regulator structure, underpinning Algorithm 2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The source for observability indices, observer canonical forms, the PBH test, and polynomial matrix descriptions used in Theorems 3 through 5."},{"cited_title":"Adaptive observers with exponential rate of conver- gence,","cited_arxiv_id":null,"evidence_quote":"The adaptive-observer filter structure that motivates the SISO tuning (73) and its MIMO generalization."},{"cited_title":"Notes on data-driven output-feedback control of linear MIMO systems,","cited_arxiv_id":null,"evidence_quote":"Discrete-time MIMO output-feedback controller synthesis where observability indices play the same role, indicating the intended generalization."}],"review_version":1}