Output behavior equivalence and simultaneous subspace identification of systems and faults

Gabriel de Albuquerque Gleizer

REVIEW 3 major objections 5 minor 30 references

Output behavior equivalence and simultaneous subspace identification of systems and faults

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict Real idea, real gap: the consistency theorem needs two-sided fault-input uncorrelatedness, but the set-valued fault recovery is worth engaging. read the letter →

arxiv 2505.13294 v1 pith:OJKRAQT4 submitted 2025-05-19 eess.SY cs.SY

Gabriel de Albuquerque Gleizer This is my paper · ORCID

classification eess.SYcs.SY

keywords systemidentificationfaultsubspacemethodsPI-MOESPoutputbehaviorequivalenceunknowninputreconstructionminimaldimensionlineartime-invariantsystems

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

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 $P$.

What carries the argument

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.

What would settle it

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 $n_v$.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

Colleague,

The 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.

That 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.

The 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.

Who 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Section III (Proposition 1)] 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 IV-C (Theorem 3)] 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 IV-C (Proposition 3)] 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.

minor comments (5)

[Introduction] The sentence 'Technically, we considersimultaneous identification' is missing a space; it should read 'we consider simultaneous identification'.
[Section II-B] 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 IV-B] 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).
[Theorem 1] The proof says 'By induction on Lemma 1' but does not spell out the induction; adding a short induction statement would improve readability.
[Figure 1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: system matrices are estimated by standard PI-MOESP and fault matrices are the exact solution set of a linear nullspace problem.

full rationale

The derivation chain is self-contained at the level of the claimed results. Proposition 1 is an application of the external standard PI-MOESP consistency result ([19], [1]) under Assumptions 1-3; the fault enters only through an additive residual and no parameter is fitted from the target output-behavior set. Propositions 2-3 and Theorem 3 are exact algebraic consequences of the residual Hankel equation R_s = [O_s T^f_s][X_s; V_s], with the fault matrices obtained as a basis of the nullspace of the Toeplitz constraints (11). Theorem 3's characterization is the solution set of those constraints, not an independently 'predicted' quantity, and the appeal to the rank formula of Theorem 2 is to an external result ([29], by Sanjeevini and Bernstein, not the present author). There are no fitted inputs renamed as predictions and no load-bearing self-citations. One caveat is flagged because the text asserts a step without full justification: in Section III the paper states 'With Assumptions 1, 2, we have that lim_{N→∞} (1/N) V_{s,N} U_{s,N}^T = 0', but Assumption 2 only constrains correlations u(k-l)v(k)^T for l>=0, i.e., current fault against past inputs, so the upper-triangular blocks of V_{s,N}U_{s,N}^T are not directly covered. This is an assumption-sufficiency gap in the proof of Proposition 1, not a circularity: the identified A,B,C,D and the fault set are not defined in terms of the target result.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central claims rest on four explicit assumptions (1-4) on input, fault, noise, and the residual system, plus standard minimality assumptions and an external rank formula from [29]. There are no invented physical entities. The only hand-chosen parameters are the window size s and a singular-value threshold in the noisy pipeline.

free parameters (2)

singular value threshold for nv selection = not specified, user-tuned
In the noisy-data part of Section V, the rank test of Proposition 3 is replaced by visual/manual thresholding of singular values (Fig. 1) to determine the fault dimension nv.
window size s = s=5 in the example
The algorithms require s >= max(l,nx); s is a design parameter of the subspace method, not fitted to data, but it must be chosen by the user.

assumptions (7)

domain assumption Assumption 1: input u is zero-mean wide-sense ergodic with full-rank block Hankel covariance
Experiment design assumption stated in Section II-B; used for consistency of PI-MOESP.
domain assumption Assumption 2: fault v is asymptotically uncorrelated with past inputs
Stated in Section II-B; this is the load-bearing premise that makes the fault act like colored noise from the perspective of past-input instruments.
domain assumption Assumption 3: noise w is WSS zero-mean and uncorrelated with input
Standard noise assumption in Section II-B.
domain assumption Assumption 4: residual system is l-delay left-invertible, (A,C) observable, [X;V] full row rank, s >= max(l,nx-1)
Stated in Section IV-C; needed for the rank formulas and for the nullspace construction of Theorem 3.
domain assumption A is Schur stable
Assumed in the system model (1); ensures boundedness of the residual filter.
domain assumption (A,C) observable and (A,B) controllable
Standard minimality assumptions in Section II-B.
standard math Rank formulas for block Toeplitz matrices (Theorem 2, adapted from [29], Theorems III.4 and IV.8)
External result used to derive Proposition 3; the paper does not prove it.

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Pith. "Pith review of Output behavior equivalence and simultaneous subspace identification of systems and faults." pith.science (2026). https://pith.science/paper/OJKRAQT4

@misc{pith2026250513294,
  author       = {Pith},
  title        = {Pith review of: Output behavior equivalence and simultaneous subspace identification of systems and faults},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OJKRAQT4}},
  note         = {Machine review of arXiv:2505.13294}
}

read the original abstract

We address the problem of identifying a system subject to additive faults, while simultaneously reconstructing the fault signal via subspace methods. We do not require nominal data for the identification, neither do we impose any assumption on the class of faults, e.g., sensor or actuator faults. We show that, under mild assumptions on the fault signal, standard PI-MOESP can recover the system matrices associated to the input-output subsystem. Then we introduce the concept of output behavior equivalence, which characterizes systems with the same output behavior set, and present a method to establish this equivalence from system matrices. Finally, we show how to estimate from data the complete set of fault matrices for which there exist a fault signal with minimal dimension that explains the data.

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

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