{"id":"07495d8f-3a28-4c58-a924-b741b3b5e91e","arxiv_id":"2505.13294","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A subspace identification method recovers nominal system matrices from faulty data and characterizes all fault matrices yielding the same output behavior, using only input-output data.","lead":"This paper shows how to identify the nominal input-output dynamics of a linear system from data corrupted by unknown additive faults, and then to reconstruct the set of fault channels that explains the data. The key new tool is a notion of output behavior equivalence that characterizes when two fault models generate the same output trajectories.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption 2 is one-sided and does not imply the zero block-Hankel cross-correlation used in the proof of Proposition 1, so an exogeneity condition on the fault is missing.","rationale":"The reader's weakest assumption is Assumption 2, and my concern also targets Assumption 2, so there is partial agreement. However, the reader framed the issue as excluding faults with a linear feedback component, which the paper itself acknowledges. I identify a sharper technical gap: the assumption as written is one-sided and does not imply the full block-Hankel cross-correlation V_{s,N}U_{s,N}^T=0 used in the proof, and it admits noncausal fault signals that break identifiability of A,B,C,D. This is load-bearing because Proposition 1 is the foundation for the residual construction R=Y−TU and therefore for the minimal fault dimension and equivalence results in Section IV. The fix is mild—strengthen Assumption 2 to full exogeneity of v with respect to u (or state that u is chosen independently of the entire fault process). The main mathematical machinery, including Proposition 3 and Theorem 3, appears sound under the intended interpretation. The reader's CONDITIONAL verdict is appropriate; my concern does not move the verdict, but it sharpens the condition that the paper must state. The concrete test settles whether the proof's cross-moment claim is actually derivable, and whether Proposition 1 fails under the stated assumption.","tokens_in":12702,"tokens_out":33617,"duration_ms":343518,"concrete_test":"Independently re-derive the implication in Section III: with v(k)=u(k+1) and u zero-mean i.i.d., verify that Assumptions 1–2 hold while the (1,2) block of (1/N)V_{2,N}U_{2,N}^T tends to E[u^2]≠0. Then run PI-MOESP on the resulting y(k)=u(k-1)+u(k) and compare the estimated D with the nominal D=0; a non-negligible D estimate demonstrates that the nominal A,B,C,D are not identified, so the theorem requires an additional exogeneity condition, e.g., two-sided asymptotic uncorrelatedness of v with u.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 1's proof invokes the claim in Section III that Assumptions 1 and 2 imply lim_N (1/N) V_{s,N} U_{s,N}^T = 0. Assumption 2 only constrains u(k-l)v(k)^T for l≥0, i.e., the current fault against past inputs. The block (i,j) with j>i of V_{s,N}U_{s,N}^T contains (1/N)Σ_k v(k)u(k+j-i)^T, a correlation of the fault with future inputs, which Assumption 2 does not cover. Concretely, take u zero-mean i.i.d. and v(k)=u(k+1). Then for every l≥0, lim_N (1/N)Σ u(k-l)v(k)=0, so Assumption 2 holds, yet the upper-triangular block of VU^T is nonzero. Moreover, the same input-output data are generated by a different causal realization (absorb v(k)=u(k+1) into D and B), so the nominal A,B,C,D are not identifiable without an additional exogeneity or causality condition on v. The paper's informal remark that v has no linear feedback component on y, x, or u is not equivalent to the stated one-sided uncorrelatedness and does not rule out noncausal dependence. Because the residual R=Y−TU and all subsequent fault-set recovery in Section IV inherit Proposition 1, the central claim rests on an under-specified assumption.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers simultaneous identification of an LTI system x(k+1)=Ax(k)+Bu(k)+Fv(k), y(k)=Cx(k)+Du(k)+Gv(k)+w(k) from input--output data generated under faults, without requiring a nominal fault-free dataset. The author first argues that PI-MOESP consistently estimates A,B,C,D under Assumptions 1--3. Then, assuming these matrices are known and the noise-free residual R=Y-TU is available, the paper derives a rank formula n_v=rank(R_{s+1})-rank(R_s) based on a theorem on transmission zeros, and characterizes the set of all fault matrices (F,G) that are output-behaviorally equivalent to the true one as {(Rhat F P, Rhat G P)}. A numerical example and a Monte Carlo study illustrate the method, and the author provides reproducible code.","tokens_in":12990,"tokens_out":19909,"duration_ms":204762,"significance":"If the consistency proof is repaired, the proposed framework is a useful practical contribution: it avoids the need for nominal data, does not commit to a structural fault class, and gives a constructive way to estimate the minimal fault dimension and the equivalence class of fault matrices. The rank-gap estimator is simple and the numerical evidence is encouraging. The author ships code for reproduction, which is a strength. The main theoretical bottleneck is the exogeneity condition behind Proposition 1; the subsequent results are interesting but rest on terse adaptations of external rank formulas and on an unproved genericity claim.","major_comments":[{"comment":"The consistency proof uses the assertion that Assumptions 1 and 2 imply lim_{N→∞} (1/N) V_{s,N} U_{s,N}^T = 0. Assumption 2 only requires lim (1/N) Σ_k u(k-l) v(k)^T = 0 for all l ≥ 0, i.e., a vanishing correlation between the current fault and past inputs. For the block (i,j) of V_{s,N} U_{s,N}^T with j>i, the sum contains terms v(k) u(k+j-i)^T, which is a correlation of the current fault with future inputs and is not covered by Assumption 2. Concretely, let u be zero-mean i.i.d. and set v(k)=u(k+1); then Assumption 2 holds for every l ≥ 0, while the (1,2) block of V_{s,N} U_{s,N}^T converges to a nonzero matrix. Hence the stated implication is false and the proof of Proposition 1 is incomplete. Since Section IV uses the identified A,B,C,D to form the residual R=Y-TU, this gap undermines the central claim of simultaneous identification. The assumption should be replaced by a two-sided exogeneity condition, e.g., lim (1/N) Σ u(k+l) v(k)^T = 0 for all l ∈ Z, or by an explicit causality/independence assumption on the fault relative to the whole input process, and the proof of Proposition 1 should be revised accordingly.","section":"Section III (Proposition 1)"},{"comment":"Theorem 3(ii) states that for almost any full-column-rank P, the system (A, Rhat F P, C, Rhat G P) is output behaviorally equivalent to the original system, but the proof only says that this follows because the column space is the same for one such P. This is a genuine genericity claim, especially when n_z>n_v (as in the numerical example, where the computed Rhat F,Rhat G has two columns while n_v=1). The matrix [O_s, Rhat T_s (I_s⊗P)] can drop rank or change its column space for non-generic P, so a proof is needed that for all P outside an algebraic set the column space agrees with that of [O_s, T_s]. As written, this load-bearing step is asserted rather than established, and the 'exactly all' part of the characterization is not fully proven.","section":"Section IV-C (Theorem 3)"},{"comment":"Proposition 3 applies Theorem 2, which is stated for a minimal system, to the residual system (A,F,C,G), yet Assumption 4 only requires (A,C) observable and the system left-invertible; it does not require controllability of (A,F). The numerical example even states that (A,F) is not controllable. If the rank formula rank([O_s,T_s])=n_x+s n_v-ζ indeed requires minimality, the formula used for n_v may fail for non-minimal residual systems. The manuscript should either add controllability of (A,F) to Assumption 4 or prove that the adapted theorem remains valid without it; otherwise the estimate of the minimal fault dimension is not supported by the stated hypotheses.","section":"Section IV-C (Proposition 3)"}],"minor_comments":[{"comment":"The sentence 'Technically, we considersimultaneous identification' is missing a space; it should read 'we consider simultaneous identification'.","section":"Introduction"},{"comment":"Assumption 1 calls the input 'zero-mean wide-sense ergodic' but the formal condition only gives a mean limit and a full-rank covariance limit; the terminology is nonstandard and should be aligned with the displayed conditions.","section":"Section II-B"},{"comment":"The notation H(q)_N for the minimal polynomial basis of the left null space is introduced without a definition; please define it explicitly before (8b).","section":"Section IV-B"},{"comment":"The proof says 'By induction on Lemma 1' but does not spell out the induction; adding a short induction statement would improve readability.","section":"Theorem 1"},{"comment":"The caption says 'only the seventh singular value of R5 is negligible compared to that of R6'; it would be clearer to state explicitly that this indicates rank(R6)-rank(R5)=1, i.e., n_v=1.","section":"Figure 1"}],"recommendation":"major_revision","confidential_remarks":"The Assumption 2 issue is substantive but fixable by strengthening the exogeneity condition to a two-sided one; the other two major concerns require either added assumptions or completed proofs. The paper is a compact note with a promising idea, and I would be willing to see a revised version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper's main identification theorem rests on an assumption that doesn't do the work it claims. The stress-test example v(k)=u(k+1) satisfies Assumption 2 (current fault uncorrelated with past inputs) yet produces a nonzero upper-triangular block in V_s U_s^T, so the PI-MOESP consistency proof fails precisely where it needs closed-loop/PI exogeneity. That is a load-bearing gap: Proposition 1 feeds the residual construction and everything in Section IV. The paper should require two-sided uncorrelatedness (or a causality condition) between fault and input, and then the PI-MOESP argument would go through.\n\nThat said, the paper is not a throwaway. The output-behavior-equivalence notion (Definition 1, Theorem 1) is a clean way to say what fault systems are indistinguishable from data, and Theorem 3's characterization of all fault matrices compatible with a given residual is genuinely new, as is Proposition 3's rank-difference formula for the minimal fault dimension. The author also ships code and does a Monte Carlo study with transmission zeros, and is candid about the cases where it fails. The numerical example showing the nuclear-norm relaxation returning a rank-2 solution where the exact method returns the correct set is a good honest comparison.\n\nThe softer spots are what you'd expect from a 6-page note: the genericity claim in Theorem 3 ('for almost any P') is not formalized, the threshold for choosing n_v from singular-value gaps is hand-tuned, and the proof of Lemma 2 in the appendix has a minor rank argument that is fine but terse. None of these are fatal by themselves; they are fixable with a longer version.\n\nWho is this for? People working in data-driven fault diagnosis and subspace identification. It deserves a serious referee, but the referee should insist on fixing Assumption 2 and making the genericity argument concrete. I wouldn't cite the consistency claim in its present form, but I would cite the behavioral equivalence and set-recovery results after the revision.","headline":"Real idea, real gap: the consistency theorem needs two-sided fault-input uncorrelatedness, but the set-valued fault recovery is worth engaging.","tokens_in":13505,"tokens_out":2878,"would_cite":false,"duration_ms":29909,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that standard PI-MOESP subspace identification consistently recovers the input-output matrices of an LTI system from data corrupted by an additive fault, and that the minimal fault dimension is the rank difference…","keywords":["system identification","fault identification","subspace methods","PI-MOESP","output behavior equivalence","unknown input reconstruction","minimal fault dimension","linear time-invariant systems"],"falsifier":"Simulate a stable LTI system with a known fault $v(k) = K u(k-1)$, violating Assumption 2, and apply PI-MOESP to the resulting $(u,y)$: a biased estimate of $(A,B,C,D)$ confirms the assumption is necessary, while an unbiased estimate would disprove Proposition 1's stated scope. In exact arithmetic with Assumption 4 satisfied, directly check whether $\\operatorname{rank}(R_{s+1}) - \\operatorname{rank}(R_s)$ equals the true fault dimension $n_v$.","tokens_in":12482,"feed_emoji":"⚙️","tokens_out":9574,"duration_ms":94423,"temperature":0.7,"pith_summary":"System identification usually wants clean data, but real systems are often already faulty. The paper asks what can be recovered from input-output data of a stable LTI system subject to an additive, unmodeled fault signal, without a fault-free experiment and without committing to a fault class such as sensor, actuator, or load. It shows that standard PI-MOESP gives consistent estimates of the nominal matrices $A, B, C, D$ when the fault is asymptotically uncorrelated with past inputs, and that the fault subsystem can then be identified from the same data. The minimal fault dimension is not guessed: under a left-invertibility condition it is exactly $\\operatorname{rank}(R_{s+1}) - \\operatorname{rank}(R_s)$. The paper also defines output behavior equivalence and proves that every set of fault matrices that produces the same output behavior is exactly parameterized by $(\\hat F P, \\hat G P)$ for full column rank $P$.","feed_headline":"Faults do not stop system identification","feed_subtitle":"PI-MOESP identifies the system from faulty data; a rank gap then sizes up the fault space.","key_machinery":"Three pieces carry the argument. First, PI-MOESP, a projection-instrumental-variable subspace identification algorithm, is applied to a reformulation in which the fault enters as an output residual $r(k)$ generated by $(A, F, C, G)$; because $u$ and $v$ are asymptotically uncorrelated, past inputs act as valid instruments and $A, B, C, D$ come out consistently. Second, the output behavior set is defined through the kernel representation $\\mathcal{B} = \\{ r : N(q) r = 0 \\}$, and Theorem 1 shows that equality of restricted behaviors on a window of length $n_x$ implies full output behavior equivalence. Third, the load-bearing identity is $n_v = \\operatorname{rank}(R_{s+1}) - \\operatorname{rank}(R_s)$, obtained by combining the transmission-zero counting formula of Theorem 2 with Assumption 4; this converts the unknown fault dimension into a rank difference. The constructive core is the nullspace problem (11): its maximal solution produces the basis matrices $(\\hat F, \\hat G)$ from which every behaviorally equivalent fault pair is generated.","core_discovery":"The central discovery is that simultaneous identification of a system and its additive faults is possible without first removing the fault or classifying it. With Assumptions 1-3, the fault residual behaves like a colored noise component that is uncorrelated with past inputs, so the standard PI-MOESP subspace method consistently estimates $A, B, C, D$ (Proposition 1). For the fault subsystem, the residual Hankel matrix obeys $R_s = [O_s\\ T_s^f] [X_s; V_s]$, and under Assumption 4 its rank is $n_x + s n_v - \\zeta$, where $\\zeta$ is the number of transmission zeros from fault to output. Therefore $n_v = \\operatorname{rank}(R_{s+1}) - \\operatorname{rank}(R_s)$ (Proposition 3). The remaining ambiguity in the fault matrices is fully characterized: after solving a linear nullspace problem, the set of all $(F,G)$ output behaviorally equivalent to the true pair is exactly $\\{(\\hat F P, \\hat G P) : P \\text{ full column rank}\\}$ (Theorem 3). This reduces fault identification to linear algebra once the nominal system is known.","pith_inferences":["The parametrization by $P$ suggests using the equivalence class as an uncertainty set: any controller or diagnosis rule that depends only on output behavior is valid across all matrices in the class, which could simplify robust fault-tolerant design.","A natural testable extension is to monitor the rank gap in real time: under Assumption 4, the index at which $\\operatorname{rank}(R_{s+1}) - \\operatorname{rank}(R_s)$ stabilizes could serve as an online detector of fault onset without a fault-free baseline.","The output behavior equivalence notion could be pushed toward a canonical form for fault matrices, so that each equivalence class has a distinguished representative with desirable numerical or structural properties."],"forward_implications":["No fault-free calibration dataset is needed: the nominal system and the fault structure can be learned from the same faulty input-output record.","The minimal fault dimension is observable from the data as a rank gap, so no prior commitment to sensor, actuator, or load fault classes is required.","The full set of $(F,G)$ pairs that explain the data is known, so fault diagnosis and reconstruction can distinguish what is identifiable from what is fundamentally ambiguous.","When the fault has no direct feedthrough to the output ($G=0$), the fault matrices and trajectory can be recovered exactly up to a basis choice.","The assumptions admit faults that are not wide-sense stationary, including time-varying or adversarial signals, as long as they stay uncorrelated with past inputs."],"supporting_citations":[{"why":"Supplies the PI-MOESP algorithm whose consistency under faulty data is the paper's first contribution.","marker":"[19]"},{"why":"Provides the subspace-identification setup and the PI-MOESP description that the consistency argument builds on.","marker":"[1]"},{"why":"The prior simultaneous identification-and-input-reconstruction method whose strong-input-observability assumption this paper removes.","marker":"[14]"},{"why":"Provides the zero-counting rank formula for Toeplitz observability matrices, used in Theorem 2 and Proposition 3.","marker":"[29]"},{"why":"Defines $l$-delay left invertibility and invariant zeros, which delimit when fault reconstruction is unique.","marker":"[28]"},{"why":"Supplies the behavioral kernel representation that underlies the notion of output behavior equivalence.","marker":"[25]"},{"why":"Documents the non-uniqueness caused by linear feedback in faults, motivating Assumption 2.","marker":"[20]"}],"fun_headline_variants":["Faulty data? Identify system and fault subspace","One subspace method recovers system and faults","No clean data needed for system and fault ID","System and fault ID without fault-free data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Assumption 2: the fault signal must be asymptotically uncorrelated with past inputs, so faults that contain a linear feedback component (for example, a stuck actuator whose effect depends on the current input) are excluded; if this fails, the identified system matrices are biased and the fault identification built on them is invalid.","fun_headline_variants_meta":{"raw":{"variants":["Faulty data? Identify system and fault subspace","One subspace method recovers system and faults","No clean data needed for system and fault ID","System and fault ID without fault-free data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001139,"raw_usage":{"total_tokens":4705,"prompt_tokens":901,"completion_tokens":3804,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":517,"completion_tokens_details":{"reasoning_tokens":3746}},"tokens_in":517,"tokens_out":3804,"duration_ms":33508,"temperature":1.0,"reasoning_tokens":3746,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:17:29.279842+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a stable LTI system with a known fault $v(k) = K u(k-1)$, violating Assumption 2, and apply PI-MOESP to the resulting $(u,y)$: a biased estimate of $(A,B,C,D)$ confirms the assumption is necessary, while an unbiased estimate would disprove Proposition 1's stated scope. In exact arithmetic with Assumption 4 satisfied, directly check whether $\\operatorname{rank}(R_{s+1}) - \\operatorname{rank}(R_s)$ equals the true fault dimension $n_v$.","supporting_citations":[{"cited_title":"Subspace model identification part 3. analysis of the ordinary output-error state-space model identification algorithm,","cited_arxiv_id":null,"evidence_quote":"Supplies the PI-MOESP algorithm whose consistency under faulty data is the paper's first contribution."},{"cited_title":"Verhaegen and V","cited_arxiv_id":null,"evidence_quote":"Provides the subspace-identification setup and the PI-MOESP description that the consistency argument builds on."},{"cited_title":"A subspace algorithm for simultaneous identification and input reconstruction,","cited_arxiv_id":null,"evidence_quote":"The prior simultaneous identification-and-input-reconstruction method whose strong-input-observability assumption this paper removes."},{"cited_title":"Counting zeros using observability and block toeplitz matrices,","cited_arxiv_id":null,"evidence_quote":"Provides the zero-counting rank formula for Toeplitz observability matrices, used in Theorem 2 and Proposition 3."},{"cited_title":"L-delay input and initial-state reconstruction for discrete- time linear systems,","cited_arxiv_id":null,"evidence_quote":"Defines $l$-delay left invertibility and invariant zeros, which delimit when fault reconstruction is unique."},{"cited_title":"Verhaegen, C","cited_arxiv_id":null,"evidence_quote":"Documents the non-uniqueness caused by linear feedback in faults, motivating Assumption 2."}],"review_version":1}