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REVIEW 3 major objections 6 minor 9 references

Observer-Based Target Control for Mismatched Time-Delay Systems

T0 review · 3 major / 6 minor · reviewed 2026-07-09 · glm-5.2

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Pith's one-line read Stabilize only what matters, even when every channel lags

desk verdict Sound framework with a real but likely fillable proof gap read the letter →

arxiv 2607.07036 v1 pith:INPMY6NL submitted 2026-07-08 eess.SY cs.SY

Hieu Trinh This is my paper · ORCID
classification eess.SYcs.SY
keywords targetcontrolmismatchedobserver-basedoutputinputlatencieslinear
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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,

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

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.

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 (3)
  1. §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
  2. §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.
  3. §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.
minor comments (6)
  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No genuine circularity; self-citations provide methodological tools (LMI conditions, observer gain procedures) rather than defining the central result in terms of itself

full rationale

The paper's central claim — that projecting full-order dynamics onto the row space of Fo yields a reduced-order subsystem whose closed-loop spectrum is preserved in the full-order system — is derived self-containedly for the τu=τx case via Lemmas 1-2 (cited from [1], Fernando & Darouach, an external work) and the algebraic identity N̄_o Fo - Fo Ā = 0 (Eq. 16). No step in this derivation reduces to its own inputs by construction. The rank condition (7) is a structural assumption, not a fitted parameter renamed as a prediction. The six self-cited companion preprints [2,3,5,6,7,8] supply methodological tools — LMI feasibility conditions for gain computation (Lemma 13 in [3]), observer gain design procedures ([8]), and dual-observer architecture ([2]) — rather than load-bearing theorems that define the central result. The spectral inclusion for the mismatched-delay case (τx > τu, control law (5)) is asserted without proof ('this derivation is omitted here for brevity'), which is a completeness/correctness gap but not circularity: the claim is not defined in terms of itself, nor is a fitted parameter presented as a prediction. The numerical examples validate specific instances but do not constitute the general proof. Score 2 reflects the density of self-citations to unrefereed preprints for critical design tools, which creates a dependency chain, but the core algebraic framework and spectral inclusion result are independently grounded.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities or postulated objects. The free parameters are standard control gains determined by LMI optimization. The axioms are standard structural assumptions for time-delay systems, except for the reliance on companion preprint lemmas.

free parameters (3)
  • Z_tau_u = Example 1: [[5.2354, -7.7006], [-3.7586, 4.0337]]
    Control gain matrices computed via LMI feasibility (Lemma 13 in [3]) for specific delay values.
  • Z_tau_x = Example 1: [[4.2524, -5.1863], [-2.8542, 2.6962]]
    Control gain matrices computed via LMI feasibility for specific delay values.
  • lambda = 1
    LMI tuning parameter set to 1 in examples to ensure feasibility.
assumptions (4)
  • domain assumption Rank condition (7): rank([Fo*A; Fo]) = rank([Fo*Ad; Fo]) = rank(Fo)
    Required for the reduced-order model derivation in Section II to decouple target dynamics from the full state.
  • domain assumption tau_x > tau_u
    Assumed without loss of generality for the main control law design (Section I); the alternative case is deferred to Remark 2.
  • domain assumption B has full column rank and C has full row rank
    Standard assumptions stated in Section I for observer/controller design.
  • ad hoc to paper Lemma 13 in [3] provides sufficient LMI conditions for stability
    The control gain synthesis relies entirely on the feasibility of an LMI formulated in a companion preprint.

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Cite this review

Pith. "Pith review of Observer-Based Target Control for Mismatched Time-Delay Systems." pith.science (2026). https://pith.science/paper/INPMY6NL

@misc{pith2026260707036,
  author       = {Pith},
  title        = {Pith review of: Observer-Based Target Control for Mismatched Time-Delay Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/INPMY6NL}},
  note         = {Machine review of arXiv:2607.07036}
}
abstract

This paper addresses observer-based target control for linear time-delay systems subject to simultaneous, mismatched input and output latencies. While full-state regulation is often conservative and computationally intensive, practical engineering objectives typically require controlling only specific linear combinations of states, or target outputs. To overcome the challenges posed by these asymmetric, dual-channel delays, we propose a reduced-order modeling framework inspired by the structural philosophy of Fernando and Darouach \cite{Fernando2025}. By projecting the high-dimensional plant dynamics onto the row space of the target output matrix $F_o$, the controller focuses strictly on the lower-dimensional target subspace. Based on this projection, an observer-based control scheme is developed to ensure precise target stabilization despite the simultaneous, mismatched input, state, and output latencies.

Figures

Figures reproduced from arXiv: 2607.07036 by the authors.

Figure 1
Figure 1. displays the trajectories of zo1(t) and zo2(t), which clearly converge to 0 as t → ∞. Although the closed-loop system (17) is inherently unstable, the target output vector zo(t) is successfully stabilized by the delayed target output controller u(t − 0.5) = Zτu zo(t − 0.5). 0 2 4 6 8 10 12 14 16 18 20 Time (seconds) -20 -15 -10 -5 0 5 10 15 z o1(t) z o2(t) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Trajectories of zo1(t) and zo2(t) under the proposed target output control law u(t − 0.5) = Zτu zo(t − 0.5) + Zτx zo(t − 1) Remark 1 (Rank Condition Fulfillment via Augmentation): When the algebraic rank condition (7) does not hold, we can construct an augmented functional matrix F¯ o of the form F¯ o =  Fo R  by choosing an auxiliary matrix R ∈ R (q−m)×n, where m < q ≤ n, such that F¯ o satisfies the rank conditi… view at source ↗
Figure 3
Figure 3. Trajectories of zo(t) and za(t) under the proposed target output control law u(t − 0.5) = Z¯τu z¯o(t − 0.5) + Z¯τx z¯o(t − 1) where α = τu +h, and Zτu , Zh ∈ R r×m are the control gains to be designed such that the target output vector zo(t) → 0 asymptotically. Substituting (23) and (24) into (10) yields the following closed-loop time-delay systems: z˙o(t) = Nozo(t) + Nodzo(t − τx) + BoZτu zo(t − τu), (25) z˙o(t) = … view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Trajectories of zo(t): Observer-based control The remainder of this paper explores an alternative approach based on [5]. In this method, the delayed control law (5) is estimated using the reduced-order time-delay subsystem (10), whose output forms a subset of the overa…
Figure 5
Figure 5. Figure 5: Trajectories of zo1 (t) and zo2 (t): Observer-based control IV. CONCLUSION Together, this paper and the companion preprints [2], [3], [5], [6], [7], [8] constitute a multi-part study detailing distinct advancements within time-delay systems theory. Specifically, this b…

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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    Existence and design of tar get output controllers

    T. Fernando and M. Darouach, “Existence and design of tar get output controllers”, IEEE Trans. Autom. Contr ., vol. 70, no. 9, pp. 6104-6110, 2025

    work page 2025

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    On Time-Delay Compensators for Delayed-Output Systems

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    work page Pith review arXiv doi:10.48550/arxiv.2606.10308 2026

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    Delayed Functional Observers for the Realization of Generalized Delayed Control Laws

    H. Trinh, “Delayed functional observers for the realiza - tion of generalized delayed control laws”, Preprint at https://doi.org/10.48550/arXiv.2606.22761 (2026)

    work page Pith review arXiv doi:10.48550/arxiv.2606.22761 2026

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    Time-Delay Compensators for Linear Systems with Delayed Output Measurements

    H. Trinh, P . T. Nam and T. N. Nguyen, “Time-Delay compensa tors for linear systems with delayed output measurements”, Prep rint at https://doi.org/10.48550/arXiv.2604.17434 (2026)

    work page Pith review arXiv doi:10.48550/arxiv.2604.17434 2026

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    Reliably computing all character istic roots of delay differential equations in a given right half plane usi ng a spectral method,

    Z. Wu and W. Michiels, “Reliably computing all character istic roots of delay differential equations in a given right half plane usi ng a spectral method,” Journal of Computational and Applied Mathematics , vol. 236, no. 9, pp. 2499-2514, 2012

    work page 2012

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    Observer-Based Control of Linear Systems with Mismatched Input and Output Delays

    H. Trinh, P . T. Nam and T. N. Nguyen, “Observer-based cont rol of linear systems with mismatched input and output delays”, Preprint at https://doi.org/10.48550/arXiv.2606.03081 (2026)

    work page Pith review arXiv doi:10.48550/arxiv.2606.03081 2026

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    Existence and desig n of functional observers for time-delay systems with delayed o utput mea- surements,

    H. Trinh, P . T. Nam, and T. Fernando, “Existence and desig n of functional observers for time-delay systems with delayed o utput mea- surements,” arXiv preprint arXiv:2603.09395, 2026

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    work page doi:10.48550/arxiv.2606 2026

Show all 9 references
  1. [9]

    Trinh, V

    H. Trinh, V . T. Huynh, S. Y u and T. Fernando, Unknown Inputs Estimation in Linear Time-Delay Systems Using Generalized Functional Observers. Springer Cham, 2026. 9

    2026

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