{"id":"3d380838-ac52-412f-9bd9-d5824a5b96e7","arxiv_id":"2607.07036","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":3,"one_line_summary":"A reduced-order observer-based control scheme stabilizes target outputs of linear time-delay systems with mismatched input, state, and output latencies by projecting dynamics onto the target output subspace.","lead":"This paper designs a reduced-order controller that stabilizes only specific target outputs of linear time-delay systems with mismatched input, state, and output delays. Engineers working with systems where full-state control is unnecessary or too costly may find the framework useful.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Spectral inclusion proof only covers τu=τx case; the mismatched-delay case (the paper's main contribution) is asserted without derivation.","rationale":"The reader correctly identified the rank condition as a structural dependency and the deferral to companion preprints as a verification barrier. However, the more acute load-bearing concern is internal to this paper: the spectral inclusion proof—the mathematical foundation validating the reduced-order approach—is explicitly omitted for the mismatched-delay case that constitutes the paper's primary contribution. The reader focused on the rank condition's restrictiveness and the external dependency on companion preprints. While valid, the rank condition has an augmentation workaround (Remark 1), and companion preprints are a verifiability issue rather than a correctness issue within this manuscript. The omitted proof for the mismatched case is a direct gap in the paper's own central argument. The reader's verdict of CONDITIONAL remains appropriate, but the specific condition should be tightened: the paper needs to either provide the omitted derivation or explicitly state that the mismatched-delay spectral inclusion is conjectured rather than proven. The numerical examples provide partial evidence but cannot substitute for the general proof, especially since the spectral inclusion property is sensitive to delay values and matrix structure.","tokens_in":14250,"tokens_out":635,"duration_ms":262448,"concrete_test":"Derive the spectral inclusion for the mismatched case by applying control law (5) to the full-order system (1) and to the reduced-order system (10). Show that under rank condition (7), the matrix equations (14)-(15) extend to yield σ(N_o + N_od e^{-λτx} + B_o Z_τu e^{-λτu} + B_o Z_τx e^{-λτx}) ⊆ σ(A + A_d e^{-λτx} + B Z_τu F_o e^{-λτu} + B Z_τx F_o e^{-λτx}). If this inclusion does not hold, the core framework is invalid for mismatched delays.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central claim is that target output stabilization can be achieved under simultaneous, mismatched input, state, and output delays. The spectral inclusion result (σ(N̄_o) ⊆ σ(Ā)), which validates that the reduced-order controller stabilizes the full-order system, is explicitly proven only for the simplified case τu = τx under control law (13). For the actual mismatched case τx > τu with control law (5), the text states: 'When using the control law (5) to stabilize (10), similar spectral properties can be established; this derivation is omitted here for brevity.' This is the load-bearing step: without this proof, there is no formal guarantee that the reduced-order closed-loop spectrum is preserved in the full-order dynamics when delays are mismatched. The entire framework's validity for its stated problem domain rests on an unproven assertion. While Example 1 provides numerical evidence for one specific case, the general theoretical gap remains.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper addresses observer-based target control for linear time-delay systems with simultaneous, mismatched input, state, and output delays. The core idea is to project the full-order state dynamics onto the row space of the target output matrix $F_o$, yielding a reduced-order time-delay subsystem of dimension $m$ (or $q$ after augmentation). A delayed control law is designed to stabilize this reduced subsystem, and a dual-observer architecture is proposed to estimate the control law from delayed output measurements. The paper builds on the structural philosophy of Fernando and Darouach [1] and extends the author's own companion preprints [2, 3, 5, 6, 7, 8] to the mismatched-delay setting. Four illustrative examples with numerical simulations are provided.","tokens_in":14922,"tokens_out":1549,"duration_ms":139950,"significance":"The target output control philosophy—stabilizing only a lower-dimensional functional of the state rather than the full state vector—is practically motivated and well-articulated. The spectral inclusion result (Eqs. 14–16), even though proven only for a special case, is a useful formal contribution. The dual-observer decomposition that exploits geometric alignment between $F_x$ and the row spaces of $C$ and $F_u$ to reduce observer complexity is an elegant architectural idea. The examples are concrete and include specific gain matrices, which aids reproducibility. However, the paper functions more as a framework overview than a self-contained theoretical contribution, with critical design steps deferred to companion preprints.","major_comments":[{"comment":"§II, spectral inclusion for the mismatched-delay case: The central contribution of the paper is target stabilization under mismatched delays ($τ_x > τ_u$) via control law (5). However, the spectral inclusion result $σ(N̄_o) ⊆ σ(Ā)$, which validates that the reduced-order controller stabilizes the full-order system, is explicitly proven only for the simplified case $τ_u = τ_x$ under control law (13). For the mismatched case with control law (5), the text states: 'When using the control law (5) to stabilize (10), similar spectral properties can be established; this derivation is omitted here for brevity.' This is the load-bearing step for the paper's stated contribution. Without this proof or at least a detailed sketch, there is no formal guarantee that the reduced-order closed-loop spectrum is preserved in the full-order dynamics when delays are mismatched. The authors should either (a)提供","section":null},{"comment":"§III, observer design: Theorem 1 states that the observer (29)–(30) provides asymptotic estimation if $C = 0$ and the error dynamics (32) are stable, but the actual computation of the observer gain matrices ($M_1$, $M_{1β}$, $N_1$, $N_{1τ}$, $N_{1τx}$, $G_i$, $J$, $J_1$, $J_2$) is entirely deferred to [8] and related preprints. The paper states 'the observer gains required to satisfy Theorem 1 can be computed by following a design procedure similar to the one in [8],' but does not even summarize the procedure. Since the observer is the implementation mechanism for the proposed control law, this makes the paper incomplete as a standalone contribution. At minimum, the key equations for solving the Sylvester-type conditions $C_i = 0$ and the LMI for error-system stability should be summarized, even if concisely.","section":null},{"comment":"§III, dual-observer for $z_2(t)$: The second observer, which estimates $z_2(t) = F_x x(t - τ_x)$, is described only qualitatively. The decomposition of $F_x$ into components within $row(F_u, C)$ and a residual $F̄_x$ is mentioned but the actual observer structure, its existence conditions, and its error dynamics are not presented. The text says 'The full derivation of this decomposition is omitted here for brevity.' Since the dual-observer is the proposed implementation architecture, omitting half of it leaves the reader unable to verify or reproduce the approach. Example 3 shows that $F_x$ lies entirely within $row(F_u)$, which is a special case; the general case where a residual $F̄_x$ remains is not illustrated.","section":null}],"minor_comments":[{"comment":"Eq. (6): The expansion of $F_o A(I - F_o^- F_o + F_o^- F_o)x(t)$ is algebraically correct but the intermediate step could be written more transparently to show how the $F_o A(I - F_o^- F_o)$ terms arise.","section":null},{"comment":"Remark 2, Eqs. (21)–(24): The control law (22) introduces an additional delay $h > 0$ and a gain $Z_h$, but the motivation for introducing this extra delay is not clearly explained beyond a reference to [3]. A brief sentence explaining the engineering rationale would help.","section":null},{"comment":"Example 1: The eigenvalues of the full-order closed-loop system (17) are listed as $¥{1.1896, 0.5751, -0.6112 ± j3.3712¥} ¥cup ¥{-0.4537 ± j1.5519, -0.4646, -3.6765¥}$. The first set has positive real parts, confirming that the full state is unstable while the target output is stabilized. This is a nice illustration but could be stated more explicitly in the text.","section":null},{"comment":"The paper states 'we consider the case where $τ_x > τ_u$' but then Example 1 first treats $τ_u = τ_x = 0.5$ before moving to $τ_x = 1, τ_u = 0.5$. The structure could be signposted more clearly.","section":null},{"comment":"Several references ([2], [3], [5], [6], [7], [8]) are to preprints by the same author, all dated 2026. The paper should clarify which of these have been peer-reviewed or accepted, as the theoretical framework depends on results established in these works.","section":null},{"comment":"Figure quality: The trajectory plots (Figs. 1–5) are small and the axis labels are difficult to read. Higher-resolution versions would be needed for print reproduction.","section":null}],"recommendation":"major_revision","confidential_remarks":"The paper is part of a multi-preprint series by the same author (explicitly acknowledged in the Conclusion). While this is not inherently problematic, the degree to which critical results are deferred to these companion preprints is unusual. The spectral inclusion gap for the mismatched case is the most serious concern: the paper's title and abstract promise a solution for mismatched delays, but the formal validation only covers the matched case. If the authors can provide the omitted proof (or a rigorous sketch) and summarize the observer design procedures, the paper could become a solid contribution. I would not recommend rejection at this stage, but the revisions must address the theoretical gap directly."},"author_rebuttal":null,"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper applies Fernando and Darouach's target output control philosophy to linear systems with simultaneous mismatched input, state, and output delays, paired with a dual-observer architecture. The projection idea itself is attributed to [1]; the novelty is in the combination with mismatched delays and the observer scheme. It's a legitimate extension within an established research program, not a new paradigm.","headline":"Sound framework with a real but likely fillable proof gap","tokens_in":14766,"tokens_out":1641,"would_cite":false,"duration_ms":75130,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Stabilize only what matters, even when every channel lags","keywords":[],"falsifier":"Find a physical or numerical example where the rank condition (7) holds, the reduced-order closed-loop is asymptotically stable, the spectral inclusion is verified, yet the target output of the full-order closed-loop system fails to converge to zero due to an interaction between the unregulated state components and the target subspace that the spectral-inclusion argument does not capture.","tokens_in":14504,"feed_emoji":"🎯","tokens_out":776,"duration_ms":117556,"temperature":0.7,"pith_summary":"This paper shows that for linear systems plagued by simultaneous but unequal delays in the state, control input, and sensor measurements, one need not stabilize the entire state vector. By projecting the full system dynamics onto the row space of a target output matrix Fo, the authors derive a reduced-order subsystem that evolves only in the dimensions an engineer actually cares about. A delay-compensated control law is then designed for this low-dimensional subsystem, and a dual-observer architecture reconstructs the necessary feedback signals from delayed measurements. The key structural result is that the eigenvalue spectrum of the reduced-order closed-loop system is provably embedded within the spectrum of the full-order closed-loop system, so stability verified on the surrogate guarantees stability of the target output in the real plant, even when the full plant remains unstable.","feed_headline":"Stabilize only what matters, even when every channel lags","feed_subtitle":"A projection method cuts time-delay control down to the target subspace, leaving the rest of the plant free to drift.","key_machinery":"The projection onto row(Fo) via generalized inverse Fo, the rank condition rank([FoA; Fo]) = rank([FoAd; Fo]) = rank(Fo) for exact decoupling, the spectral inclusion sigma(N_bar_o) subseteq sigma(A_bar), the augmented observability matrix construction for rank-condition relaxation, and the dual-observer architecture splitting the control-law estimation into two parallel functional observers.","core_discovery":"The central mechanism is a projection-based order reduction for time-delay systems. Premultiplying the state equation by the target matrix Fo yields an m-dimensional subsystem (where m is the number of target outputs, m < n) provided a rank condition holds: the rows of FoA and FoAd must lie in the row space of Fo. When this condition is met, the target dynamics decouple cleanly from the unregulated state components. The paper then proves that the closed-loop spectrum of this projected subsystem is a subset of the full-order closed-loop spectrum, establishing that a controller designed on the reduced model legitimately governs the target behavior of the full model. When the rank condition is,","pith_inferences":[],"forward_implications":["Control engineers can regulate specific performance variables in high-dimensional delayed plants without bearing the computational cost or conservatism of full-state observer design.","The spectral inclusion result means reduced-order stability certificates are valid for the full system, potentially simplifying certification of safety-critical delayed control loops.","The augmentation procedure (Remark 1) provides a systematic fallback: when exact projection fails, one enlarges the target subspace using observability-matrix rows until the rank condition is met, trading dimensionality for structural feasibility.","The dual-observer splitting strategy, where the second functional is reconstructed algebraically from the first when geometric conditions permit, could generalize to other observer-based architectures for systems with multiple mismatched delays."],"fun_headline_variants":["Observer-based target control survives mismatched dual-channel delays","Project onto target subspace to stabilize time-delay systems with asymmetric lags","Reduced-order observer controls only target outputs despite mismatched latencies","Target subspace projection decouples control from unregulated states under dual delays","Rank condition enables reduced-order target stabilization for mismatched time-delay plants"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The entire framework depends on an algebraic rank condition requiring that the target output matrix, multiplied by the system and delay matrices, does not generate new row directions outside the target subspace. When this condition fails, the paper proposes augmentation, but the core reduced-order decoupling structurally requires this property or its augmented equivalent.","fun_headline_variants_meta":{"raw":{"variants":["Observer-based target control survives mismatched dual-channel delays","Project onto target subspace to stabilize time-delay systems with asymmetric lags","Reduced-order observer controls only target outputs despite mismatched latencies","Target subspace projection decouples control from unregulated states under dual delays","Rank condition enables reduced-order target stabilization for mismatched time-delay plants"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":577,"prompt_tokens":488,"completion_tokens":89,"prompt_tokens_details":null},"tokens_in":488,"tokens_out":89,"duration_ms":20048,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T21:12:13.557648+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"Find a physical or numerical example where the rank condition (7) holds, the reduced-order closed-loop is asymptotically stable, the spectral inclusion is verified, yet the target output of the full-order closed-loop system fails to converge to zero due to an interaction between the unregulated state components and the target subspace that the spectral-inclusion argument does not capture.","supporting_citations":[],"review_version":1}