{"id":"519fa47f-b68a-411e-ac8e-223d37c58692","arxiv_id":"2502.04062","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For stable reachable linear systems, the state is persistently exciting exactly when a window of input shifts or derivatives is persistently exciting, with weaker partial-excitation conditions needed for multi-input systems.","lead":"This paper derives input-signal conditions that guarantee a stable linear system's state stays persistently exciting, in both discrete and continuous time. It unifies earlier sufficient-richness results by treating time shifts and derivatives as the same operation, and it shows the multi-input case only admits bracketed characterizations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2 and Lemma 4 assume full row rank for [M − \\tilde E], but in the multi-input partial-PE regime this matrix has more rows than columns, so the kernel-dimension bound is invalid and the new necessity theorems are unsupported.","rationale":"The reader's weakest_assumption identifies the same load-bearing defect as I find: the rank-nullity computation in Lemmas 2 and 4 is invalid in exactly the regime needed for the multi-input necessary conditions. I checked the dimensions directly. For Lemma 2 with d=2, n=3, d′=1, k=4, the matrix has more rows than columns, so full row rank is impossible and the formula for the kernel dimension is negative. The same is true for Lemma 4 with large N. These lemmas are not peripheral; Theorem 1(ii) and Theorem 3(ii) depend on them for the multi-input case, and the claimed characterization of sufficient-richness sets in Lemma 7 inherits the problem. The paper's sufficient conditions (Theorems 2 and 4) are mostly known or plausible, and the single-input characterization may survive, but the central new necessary conditions are not rigorously established. The numerical counterexamples are finite-horizon rank plots and cannot replace an asymptotic proof. I therefore concur with the reader's REJECT verdict; the appropriate outcome is a major revision in which the rank step is either repaired or the multi-input necessity claims are withdrawn.","tokens_in":29975,"tokens_out":7415,"duration_ms":74757,"concrete_test":"Compute the dimensions in Lemma 2 for d=2, n=3, d′=1, k=4. The stacked matrix [M − \\tilde E] from (40)–(41) has 12 rows and 10 columns, so it cannot have full row rank; the rank-nullity formula (43) gives d′′ = −2, an impossible kernel dimension. Similarly, for Lemma 4 take d=2, n=3, d′=1, N=2: the matrix has 12 rows and 7 columns, again contradicting the claimed full-row-rank assumption. If the necessity proofs cannot be repaired by replacing this step with a valid rank argument, Theorems 1 and 3 lack support.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step for the new multi-input necessary conditions is Lemma 2 (discrete time) and Lemma 4 (continuous time). In the proof of Lemma 2, the matrix [M − \\tilde E] has M ∈ R^{nd(k−n+1)×kd} and \\tilde E = I_{k−n+1}⊗E ∈ R^{nd(k−n+1)×d′(k−n+1)}. The proof asserts that [M − \\tilde E] has full row rank and then computes dim ker = kd + d′(k−n+1) − nd(k−n+1) = (k−n)(d+d′−nd)+d′. But full row rank is impossible whenever the number of rows exceeds the number of columns. This is exactly the multi-input partial-PE regime: for example, with d=2, n=3, d′=1, k=4, [M − \\tilde E] is 12×10, and the displayed formula gives a kernel dimension of −2, which is impossible. The continuous-time Lemma 4 has the same defect: rows = ndN and columns = Nd + d′(N+1); for d=2, n=3, d′=1, N=2, rows = 12 and columns = 7. Theorem 1 invokes Lemma 2 with d=m and d′ = n′ ≤ n−1, which is precisely the regime where the rank step fails whenever m>1; Theorem 3 invokes Lemma 4 in the analogous situation. Consequently, the multi-input necessity directions of Theorems 1 and 3 are unsupported as written. The single-input scalar case d=1 is not affected by this particular dimension count, and the sufficient-condition proofs do not rely on these rank-nullity claims.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies sufficient richness for linear time-invariant systems. For discrete- and continuous-time stable, reachable systems with output x or (x,u), it claims necessary and sufficient conditions on the input for the output to be persistently exciting: PE of the state forces the n-window stack (or derivative stack) of the input to be partially PE of degree n (or n+m for state-input), and full PE of that stack is sufficient. The paper also characterizes the single-input sufficiently rich set exactly and gives inclusions for multi-input systems. The main novelty claimed is the multi-input necessary direction, which relies on two lemmas (Lemma 2 and Lemma 4) asserting that the partial-PE degree of a stacked signal does not increase when the window length grows.","tokens_in":30305,"tokens_out":8219,"duration_ms":82454,"significance":"If the results were correct, the paper would unify and extend existing sufficient-richness conditions, in particular giving the first necessary conditions for multi-input systems and a clean single-input characterization. The paper is clearly written, uses a unifying notation, and includes numerical examples demonstrating tightness. However, the central new necessary conditions for multi-input systems rest on rank-nullity arguments in Lemmas 2 and 4 that are invalid: the matrices involved do not have full row rank in exactly the regime where the lemmas are applied, and the displayed kernel dimensions can be negative. The proofs of several theorem statements are also explicitly omitted or only sketched (Theorem 3(ii) is not proved at all). Consequently, the main claimed contribution is not established as written. The sufficient-condition directions and the single-input characterization appear more robust, but those are largely known or are not the paper's principal novelty.","major_comments":[{"comment":"Lemma 2 is not proved correctly. The matrix [M − \\tilde E] has dimensions nd(k−n+1) × (kd + d′(k−n+1)). The proof asserts that it has full row rank and computes dim ker = kd + d′(k−n+1) − nd(k−n+1) = (k−n)(d+d′−nd) + d′. Full row rank is impossible whenever the number of rows exceeds the number of columns, which is exactly the regime d′ ≤ d(n−1) for the multi-input applications. For example, with d=2, n=3, d′=1, k=4, the matrix is 12×10 and the displayed formula gives a kernel dimension of −2. Therefore the conclusion d′′ ≤ d′ is unsupported, and the necessity direction of Theorem 1, which invokes Lemma 2 with d=m and d′=n′≤n−1, is not established for multi-input systems.","section":"Appendix VI-B, Eq. (43)"},{"comment":"Lemma 4 has the same defect as Lemma 2. The matrix [M − \\tilde E] has ndN rows and Nd + d′(N+1) columns, and the proof asserts full row rank, leading to d′′ = N(d+d′−nd) + d′. For d=2, n=3, d′=1, N=2, the matrix is 12×7 and the claimed kernel dimension is negative. Since Theorem 3 (continuous-time necessity) invokes Lemma 4 with d=m and d′=n′≤n−1, the new continuous-time necessary conditions for multi-input systems are unsupported.","section":"Appendix VI-D, Eq. (66)"},{"comment":"Theorem 3(ii) is stated as a theorem but its proof is not given: Appendix VI-H ends with 'We omit the second statement'. Similarly, Theorem 1(ii) is only sketched (Appendix VI-F) with the final step omitted, and Theorem 4(ii) is sketched with explicit omissions (Appendix VI-I). A published theorem requires a complete or at least a rigorous proof of all its claims; an explicit omission is not acceptable in a journal submission. This is particularly serious because those second statements are part of the claimed characterization for the (x,u) output.","section":"Section III-B and Appendix VI-H"},{"comment":"The characterization of the sufficiently rich sets depends directly on the necessity and sufficiency theorems. Since Theorems 1 and 3 are not established for multi-input systems, the inclusions in Lemma 7 for the multi-input classes rest on unproved necessary directions. The single-input characterization in Lemma 6 may survive, but the paper's broader claim to characterize sufficiently rich inputs for multi-input systems is not supported by the current proofs.","section":"Section III-C, Lemmas 6 and 7"}],"minor_comments":[{"comment":"There are several typographical and terminology issues, such as 'Shur' in place of 'Schur' in Remarks 5 and 7, and the inconsistent use of Q_n(u) in place of D_n(u) in Remark 8.","section":"Throughout"},{"comment":"Two different plots are both labelled 'Figure 1' in the text; the second should be Figure 2. The captions are also very brief and do not fully explain the plotted quantities.","section":"Section IV, Figures 1 and 2"},{"comment":"The discrete-time PE definition uses a sum over τ = t, ..., t+T, but the window length should be specified consistently with the continuous-time definition; a short clarification would avoid ambiguity.","section":"Section II, Definition 5"},{"comment":"In the proof of Theorem 1(ii), the matrix dimensions of the stacked vector (u_{τ+1}, U_τ, ..., U_{τ-(K-1)n}) are not explicitly given; adding the dimensions would make the argument easier to follow.","section":"Appendix VI-F, Eq. (82)"}],"recommendation":"reject","confidential_remarks":"The paper has some merit in its unified notation, sufficient-condition proofs, and single-input characterization. However, the central new contribution—the multi-input necessary conditions—is based on a rank-nullity argument that is demonstrably false in the relevant regime, and a theorem statement is left unproved. These are load-bearing errors that cannot be fixed by local edits; the authors would need to supply a substantially different proof for the necessity directions. I recommend rejection, though I would not rule out a future major revision if the necessary conditions can be rigorously established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new necessary conditions for multi-input systems don't hold up as written. Lemmas 2 and 4 assert that [M − \\tilde E] has full row rank and then compute its kernel dimension, but in precisely the partial-PE regime the theorems rely on, that matrix has more rows than columns. For d=2, n=3, d′=1, k=4, the discrete-time matrix is 12×10; full row rank is impossible, and the displayed kernel dimension is negative. The continuous-time lemma has the same defect. This invalidates the necessity directions of Theorems 1 and 3 for multi-input systems. This is a load-bearing flaw in a central contribution, not a minor repair.\n\nWhat is genuinely new and good: the continuous-time sufficient condition with uniform bounds (Theorem 4) is a real extension beyond the finite-horizon WPE result in Rapisarda et al., and the unified shift/derivative notation is a genuine service to the literature. The paper is also honest: it explicitly says Theorem 3(ii) is not proved and the second statements of other theorems are only sketched. The single-input characterization in Lemma 6 is a clean consequence of the sufficient conditions plus known necessity. The counterexamples to the conjectures in [48] are interesting.\n\nThe numerical examples are finite-horizon rank plots over 1000 steps; they cannot establish the asymptotic PE or PPE claims, so the tightness argument is weaker than the text suggests. The counterexamples are suggestive but do not compensate for the proof gap.\n\nOn balance: the sufficient side is fine and the single-input story is solid, but the paper's headline claim—necessary and sufficient conditions for multi-input systems—is not supported. The paper still deserves a serious referee: the problem is important, the unifying framework is useful, and a corrected proof (or an honest restriction of the theorems to a domain where the rank argument is valid) would make the necessity results worth having. I would send it to review with a clear request to fix the rank-nullity step or delete the unsupported claims.","headline":"The new multi-input necessity proofs collapse on a rank-nullity step—full row rank is asserted for a matrix with more rows than columns—but the sufficient conditions and the single-input characterization are worth keeping.","tokens_in":30855,"tokens_out":3193,"would_cite":false,"duration_ms":30750,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B05","93C05","93C55","93D20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for asymptotically stable reachable linear time-invariant systems, persistent excitation of the state or state-input regressor is governed exactly by the partial-PE degree of a windowed stack of input time shifts…","keywords":["persistent excitation","sufficient richness","linear time-invariant systems","partial persistent excitation","time-shift operators","derivative operators","adaptive control","system identification"],"falsifier":"Set $d' < d(n-1)$ and choose $N$ so large that the stacked matrix $[M - \\widetilde{E}]$ has $ndN$ rows and $Nd + d'(N+1)$ columns; since $ndN > Nd + d'(N+1)$ when $d' < d(n-1)$, full row rank is impossible, so the kernel-dimension formula $d'' = N(d+d'-nd)+d'$ cannot hold. Checking this rank for one such system settles whether Theorem 3's necessity direction is backed by a valid argument.","tokens_in":29757,"feed_emoji":"📐","tokens_out":10311,"duration_ms":92181,"temperature":0.7,"pith_summary":"Persistent excitation of a regressor is the condition that guarantees uniform exponential parameter convergence in adaptive, identification, and learning schemes. This paper asks which input signals, when fed into an asymptotically stable reachable linear time-invariant system, keep the state (or state-input) regressor persistently exciting. For single-input systems the answer is exact: the input must itself be persistently exciting when stacked into windows of the last $n$ time shifts (discrete time) or derivatives up to order $n-1$ (continuous time). For multi-input systems the paper states separate necessary and sufficient conditions in terms of a new notion, partial persistent excitation of degree $n$ (or $n+m$ for the state-input regressor). The paper also unifies the discrete- and continuous-time frameworks through the analogy between time shifts and derivatives, and gives numerical counterexamples showing the conditions are tight.","feed_headline":"A finite input window decides persistent excitation","feed_subtitle":"In stable single-input systems, parameter convergence is guaranteed exactly when the recent input window spans all directions.","key_machinery":"The load-bearing object is the stacked history operator: $Q_k(u)$ collects the last $k$ time-shifted copies of the input (discrete time), and $D_k(u)$ collects the derivatives of the input up to order $k-1$ (continuous time). The new notion of partial persistent excitation (PPE) of degree $d'$ says that some surjective linear projection of a signal is persistently exciting, so the signal persistently spans only a $d'$-dimensional subspace. Lemmas 2 and 4 are the workhorse: once the PPE degree of the $n$-stack falls below the critical value $d(n-1)$, increasing the stack length cannot raise the PPE degree. That 'stacking cannot create new persistently spanned directions' property converts a lack of PPE in the input into a lack of PE in the state in the necessity directions; in the sufficiency directions, a non-PE state is shown to force the input to be nearly a static state feedback plus a low-dimensional residual, which prevents the stacked input from being PE.","core_discovery":"The paper's central claim is that, for asymptotically stable reachable LTI systems, PE of the output regressor is exactly a property of the stacked input history $Q_n(u)$ (discrete time) or $D_n(u)$ (continuous time). The necessity theorems (Theorem 1 discrete, Theorem 3 continuous) state that if $x$ is PE then $Q_n(u)$ (or $D_n(u)$) must be partially persistently exciting of degree $n$, and if $(x,u)$ is PE then $Q_{n+1}(u)$ (or $D_{n+1}(u)$) must be partially PE of degree $n+m$. The sufficiency theorems (Theorems 2 and 4) run in the opposite direction: full PE of the stacked input implies PE of the state, and full PE of the $n+1$ stack implies PE of the state-input pair. In the single-input case the necessary and sufficient conditions coincide, giving the explicit description of sufficiently rich signals in Lemma 6; in the multi-input case the set of sufficiently rich signals is bracketed between full PE and partial PE of degree $n$ in Lemma 7. Numerical examples show that using a weaker sufficient condition or a stronger necessary condition fails.","pith_inferences":["Editorial inference: if the rank issue in the continuous-time necessity lemma is repairable, the same sampling construction could yield finite-time, quantitative certificates for continuous-time systems, connecting classical PE certificates to finite-horizon data-driven ones.","Editorial inference: the gap between the two inclusions in Lemma 7 likely depends on how the input channels enter through the matrix $B$; a system-dependent invariant, probably the controllability index, may determine the minimal stack length needed for multi-input sufficient richness.","Editorial inference: the shift-derivative analogy suggests that nonlinear versions of the result, if they exist, would need a time-varying or Lipschitz replacement for Lemmas 2 and 4, where the critical stack length would depend on the system's rate of variation rather than on the state dimension alone."],"forward_implications":["For any stable reachable single-input system, an input is sufficiently rich if and only if its recent $n$-shift window (or $n$ derivatives in continuous time) is persistently exciting, so richness can be certified without knowing the system matrices.","The same condition with $n+1$ shifts or derivatives certifies persistent excitation of the full state-input regressor $(x,u)$, the regressor used in many data-driven and adaptive schemes.","In multi-input systems, full PE of $Q_n(u)$ is sufficient and PE of $x$ forces at least partial PE of degree $n$; these two conditions bracket the set of sufficiently rich inputs.","Because the same proof architecture covers discrete and continuous time, conditions previously stated separately for time shifts and for derivatives are shown to be one unified result.","Numerical examples show the conditions cannot be improved without adding knowledge of the specific system: weakening the sufficient stack condition or strengthening the necessary PPE condition produces inputs that fail to excite."],"supporting_citations":[{"why":"Introduces the partial-persistence notion (under another name) and the single-input system-independence result that underpins Lemma 6.","marker":"[8]"},{"why":"Supplies the discrete-time sufficient condition for state PE that Theorem 2 reproves and extends to the state-input case.","marker":"[44]"},{"why":"Provides the only earlier necessary condition (single-input, finite-horizon persistency) and states the conjectures the counterexamples address.","marker":"[48]"},{"why":"Gives the continuous-time finite-horizon sufficient condition that Theorem 4 extends to the classical uniform-in-time PE definition.","marker":"[51]"},{"why":"Contributes the multi-input sufficient-condition idea based on time shifts, which the new proofs build on.","marker":"[43]"},{"why":"Supplies the continuous-time PE definition and the lemma that PE of (x,u) implies PE of any surjective linear image.","marker":"[6]"},{"why":"Provides the reachability and controllability-index facts used in the multi-input tightness examples.","marker":"[54]"}],"fun_headline_variants":["Input stack decides persistent excitation in stable LTI systems","Parameter convergence pinned by recent input window only","PE for LTI reduces to richness of stacked input signal","Single-input LTI: PE exactly input richness, no more","Exact PE conditions via input history for stable LTI"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The continuous-time necessity proof assumes that a certain stacked matrix built from sampled derivatives has full row rank, but in the parameter range $d' < d(n-1)$ that matrix can have more rows than columns, so full row rank is impossible; if that rank argument cannot be repaired, the new continuous-time necessary condition loses its proof.","fun_headline_variants_meta":{"raw":{"variants":["Input stack decides persistent excitation in stable LTI systems","Parameter convergence pinned by recent input window only","PE for LTI reduces to richness of stacked input signal","Single-input LTI: PE exactly input richness, no more","Exact PE conditions via input history for stable LTI"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000166,"raw_usage":{"total_tokens":1238,"prompt_tokens":915,"completion_tokens":323,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":531,"completion_tokens_details":{"reasoning_tokens":245}},"tokens_in":531,"tokens_out":323,"duration_ms":3569,"temperature":1.0,"reasoning_tokens":245,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T23:41:33.789598+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set $d' < d(n-1)$ and choose $N$ so large that the stacked matrix $[M - \\widetilde{E}]$ has $ndN$ rows and $Nd + d'(N+1)$ columns; since $ndN > Nd + d'(N+1)$ when $d' < d(n-1)$, full row rank is impossible, so the kernel-dimension formula $d'' = N(d+d'-nd)+d'$ cannot hold. Checking this rank for one such system settles whether Theorem 3's necessity direction is backed by a valid argument.","supporting_citations":[{"cited_title":"Persistent excitati on in adaptive systems,","cited_arxiv_id":null,"evidence_quote":"Introduces the partial-persistence notion (under another name) and the single-input system-independence result that underpins Lemma 6."},{"cited_title":"Persistency of excitation, sufficient richness and parameter convergence in discrete t ime adaptive control,","cited_arxiv_id":null,"evidence_quote":"Supplies the discrete-time sufficient condition for state PE that Theorem 2 reproves and extends to the state-input case."},{"cited_title":"On the persistency of excitation,","cited_arxiv_id":null,"evidence_quote":"Provides the only earlier necessary condition (single-input, finite-horizon persistency) and states the conjectures the counterexamples address."},{"cited_title":"A persi stency of excitation condition for continuous-time system s,","cited_arxiv_id":null,"evidence_quote":"Gives the continuous-time finite-horizon sufficient condition that Theorem 4 extends to the classical uniform-in-time PE definition."},{"cited_title":"Persistence of excitation in extended least squares,","cited_arxiv_id":null,"evidence_quote":"Contributes the multi-input sufficient-condition idea based on time shifts, which the new proofs build on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the continuous-time PE definition and the lemma that PE of (x,u) implies PE of any surjective linear image."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the reachability and controllability-index facts used in the multi-input tightness examples."}],"review_version":1}