Delayed Functional Observers for the Realization of Generalized Delayed Control Laws

Hieu Trinh

arxiv: 2606.22761 · v1 · pith:BKHM56DPnew · submitted 2026-06-22 · 📡 eess.SY · cs.SY

Delayed Functional Observers for the Realization of Generalized Delayed Control Laws

Hieu Trinh This is my paper

Pith reviewed 2026-06-26 07:30 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords delayed functional observerstime-delay systemsgeneralized delayed control lawsobserver-based controlmismatched delaysinput delaysoutput delaysstabilization

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The pith

Delayed functional observers estimate generalized delayed control laws to stabilize systems with mismatched input and output time-delays.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proposes the design of delayed functional observers that asymptotically estimate a generalized delayed control law in systems with significant input and output delays. The approach enables the observer-based control scheme to stabilize the system even when the delays are mismatched. It also extends the allowable bounds for input delays. A reader would care because many practical systems involve time delays that can destabilize feedback loops if not properly handled.

Core claim

Building on the collective advancements in the literature, this paper proposes the design of delayed functional observers to asymptotically estimate a generalized delayed control law under significant input and output delays. This framework enables designers to extend the allowable bounds for input delays while ensuring that the observer-based control scheme stabilizes the system despite simultaneous mismatched input and output time-delays.

What carries the argument

Delayed functional observers that asymptotically estimate the generalized delayed control law, allowing realization of the control despite delays.

If this is right

The allowable bounds for input delays can be extended while maintaining stability.
The system remains stable despite simultaneous mismatched input and output time-delays.
Observer-based control schemes can be applied to a broader range of time-delay systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This approach may apply to real-time control in communication networks with variable delays.
Future work could test the observers in experimental setups with actual hardware delays.
Similar designs might be adapted for systems with multiple delays or nonlinear dynamics.

Load-bearing premise

The system satisfies the existence and stability conditions established in the cited prior works for the delayed functional observers and the closed-loop system.

What would settle it

A simulation or experiment where the proposed observers are applied but the closed-loop system becomes unstable when input delays exceed the newly extended bounds, or when the prior conditions are violated.

Figures

Figures reproduced from arXiv: 2606.22761 by Hieu Trinh.

**Figure 1.** Figure 1: Trajectories of x1(t) and x2(t): Generalized delayed control law utilizing the observer framework (7)-(8). Based on the design procedure [4], we obtain the following second-order functional observer implementation zˆ1(t) = 1 0 zˆaug(t), where zˆaug(t) = w1(t) + −3.4648 1.8067 y(t), ˙zˆaug(t) = 6.8658 11.6406 −3.0418 −4.8658 zˆaug(t) + −2.4800 −1.3177 1.0385 0.1960 zˆaug(t − 0.45) + 1.5227 −0.… view at source ↗

**Figure 3.** Figure 3: Trajectories of x1(t) and x2(t): Observer-based control with R = 5 0 Example 1), or inherently due to physical constraints imposed on the output measurement delay. In either case, the estimation is first carried out by using a primary observer to estimate the intermediate state functional z1(t) = F x(t − τu). This initial step can be realized via a functional observer of the form (7)-(8) with the net dela… view at source ↗

**Figure 4.** Figure 4: shows the trajectories of x1(t) and x2(t). It is clear that asymptotic stability of the closed-loop system has been achieved. 0 5 10 15 20 25 30 35 40 Time (seconds) -10 -5 0 5 10 15 x 1 (t) x 2 (t) [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: Trajectories of x1(t) and x2(t): Observer-based control with R = 5 0 , τy > h III. CONCLUSION This paper expands upon the existing literature [1], [2], [3], [4], [5] by developing a robust functional observer framework tailored for systems with significant, mismatched input and output delays. The proposed method guarantees the asymptotic estimation of a generalized delayed control law, ultimately providi… view at source ↗

read the original abstract

Building on the collective advancements in the literature \cite{trinh1, trinh2, trinhnn26, trinhnam26, trinhnam1}, this paper proposes the design of delayed functional observers to asymptotically estimate a generalized delayed control law under significant input and output delays. This framework enables designers to extend the allowable bounds for input delays while ensuring that the observer-based control scheme stabilizes the system despite simultaneous mismatched input and output time-delays.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a narrow incremental extension of the author's own prior results on delayed functional observers, with the mismatched-delay claim depending entirely on inherited conditions from those self-citations.

read the letter

The core point is that the paper takes the author's earlier designs for delayed functional observers and applies them to a generalized control law under simultaneous input and output delays, claiming this lets the closed-loop system tolerate larger input delays than before.

What is actually new is the specific observer construction aimed at estimating that generalized law while handling mismatched delays. The abstract frames the work as building directly on the cited sequence of Trinh papers, so the technical step is applying those existing existence and stability results to this setting rather than deriving fresh conditions from scratch.

The paper does address a concrete issue that comes up in some networked or transport-delay systems where input and output delays differ. If the full derivations in the manuscript simply reuse the prior LMIs or error dynamics without new coupling terms, then the extension is straightforward and the delay-bound claim follows from the old theorems.

The main soft spot is exactly the one the stress-test flags: the central claims rest on the system satisfying the conditions from trinh1, trinh2, trinhnn26, trinhnam26, and trinhnam1. If the mismatched delays introduce error-system terms or delay-dependent effects outside those earlier theorems, the asymptotic estimation and the extended input-delay bound may not hold. The abstract gives no indication that the paper re-derives or explicitly checks this, so a referee would need to verify whether the generalization is covered or requires additional analysis.

This is niche work for people already working on time-delay observers and functional observers in linear systems. A specialist might pick up the construction for a specific application, but it is not the kind of result that changes the broader approach to the area.

It is solid enough on its own terms to go to peer review rather than desk reject; the math can be checked and the delay-extension claim is in principle falsifiable. I would send it out, with the expectation that reviewers will press on how much is truly new versus inherited.

Referee Report

2 major / 0 minor

Summary. The paper proposes the design of delayed functional observers to asymptotically estimate a generalized delayed control law for linear systems subject to simultaneous mismatched input and output time-delays. Building on prior results, the framework is claimed to enable extension of allowable input delay bounds while ensuring stability of the resulting observer-based closed-loop control scheme.

Significance. If the observers provide the claimed asymptotic estimation and the closed-loop remains stable for input delays beyond those in the cited prior works, the result would offer a practical extension for handling larger delays in observer-based control of time-delay systems. The approach generalizes existing functional observer designs, but its significance is limited by the absence of new derivations or independent verification of the mismatched-delay case.

major comments (2)

[Abstract] Abstract: The central claim that the design 'enables designers to extend the allowable bounds for input delays while ensuring that the observer-based control scheme stabilizes the system' rests entirely on the system satisfying unspecified existence and stability conditions from the cited prior works (trinh1, trinh2, trinhnn26, trinhnam26, trinhnam1). No re-derivation or explicit check for new coupling terms or altered delay-dependent conditions arising from the mismatched-delay generalization is provided, which is load-bearing for the extension claim.
[Abstract] The manuscript provides no independent benchmarks, numerical examples with explicit LMI solutions, or counter-example checks to verify that the generalized observers achieve convergence and stability outside the scope of the prior theorems; this makes the extension of delay bounds unverifiable from the given text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and constructive feedback. We address each major comment below, clarifying the manuscript's reliance on prior results while proposing revisions to improve clarity and verifiability.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that the design 'enables designers to extend the allowable bounds for input delays while ensuring that the observer-based control scheme stabilizes the system' rests entirely on the system satisfying unspecified existence and stability conditions from the cited prior works (trinh1, trinh2, trinhnn26, trinhnam26, trinhnam1). No re-derivation or explicit check for new coupling terms or altered delay-dependent conditions arising from the mismatched-delay generalization is provided, which is load-bearing for the extension claim.

Authors: The contribution of this work is the generalization of the delayed functional observer structure to accommodate a generalized delayed control law under mismatched input/output delays. The existence and stability conditions remain those established in the cited references, as the mismatched delays are incorporated via the choice of the control law parameters without introducing additional coupling terms that alter the delay-dependent LMIs. We will revise the abstract and add a clarifying paragraph in Section II to explicitly note that the stability criteria are inherited from the prior theorems, provided the system matrices satisfy the referenced conditions. revision: partial
Referee: [Abstract] The manuscript provides no independent benchmarks, numerical examples with explicit LMI solutions, or counter-example checks to verify that the generalized observers achieve convergence and stability outside the scope of the prior theorems; this makes the extension of delay bounds unverifiable from the given text.

Authors: The manuscript includes a numerical example (Section IV) that solves the relevant LMIs for a system with mismatched delays and illustrates observer convergence. To directly address the verifiability concern, we will expand this example in the revision to include explicit LMI solutions, comparisons against the input delay bounds from the cited prior works, and additional simulation results confirming stability for larger delays. revision: yes

Circularity Check

1 steps flagged

Central claims inherit existence/stability conditions from self-cited prior works without re-derivation for mismatched delays

specific steps

self citation load bearing [Abstract]
"Building on the collective advancements in the literature \cite{trinh1, trinh2, trinhnn26, trinhnam26, trinhnam1}, this paper proposes the design of delayed functional observers to asymptotically estimate a generalized delayed control law under significant input and output delays. This framework enables designers to extend the allowable bounds for input delays while ensuring that the observer-based control scheme stabilizes the system despite simultaneous mismatched input and output time-delays."

The asymptotic estimation and stabilization claims are asserted only if the system satisfies the existence and stability conditions established in the cited prior works. Because those works are by the same lead author and no new derivation or explicit check for the mismatched-delay generalization appears, the central result is carried by the self-citation chain rather than by independent content in the present paper.

full rationale

The abstract states the design 'builds on' the cited works and that the framework 'enables' extended delay bounds while ensuring stabilization. The strongest claim (asymptotic estimation of the generalized delayed control law and closed-loop stability despite simultaneous mismatched delays) is conditioned on the system satisfying unspecified existence and stability conditions from trinh1, trinh2, trinhnn26, trinhnam26, trinhnam1. These citations share the lead author. No independent re-derivation, LMI verification, or external benchmark for the new mismatched-delay coupling is supplied, so the new result reduces to an incremental extension whose validity is carried by the self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; full manuscript required to populate the ledger.

pith-pipeline@v0.9.1-grok · 5589 in / 1175 out tokens · 27927 ms · 2026-06-26T07:30:42.590285+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 5 canonical work pages · 3 internal anchors

[1]

Yu, Critical Ledgers and Scale-Defect Cascades for Navier–Stokes, arXiv preprint arXiv:2606.13887 [math.AP], 2026

H. Trinh, “Delayed functional observers for output-del ayed linear sys- tems”, Preprint at https://doi.org/10.48550/arXiv.2606 .09407 (2026)

work page doi:10.48550/arxiv.2606 2026
[2]

On Time-Delay Compensators for Delayed-Output Systems

H. Trinh, “On time-delay compensators for delayed-outp ut systems”, Preprint at https://doi.org/10.48550/arXiv.2606.10308 (2026)

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.10308 2026
[3]

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 internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.03081 2026
[4]

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 internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.17434 2026
[5]

Existence and design of functional observers for time-delay systems with delayed output measu rements

H. Trinh, P . T. Nam and T. Fernando, “Existence and design of functional observers for time-delay systems with delayed output measu rements”, Preprint at https://doi.org/10.48550/arXiv.2603.09395 (2026)

work page doi:10.48550/arxiv.2603.09395 2026
[6]

Existence and design of functional observ ers

M. Darouach, “Existence and design of functional observ ers”, IEEE Trans. Autom. Contr ., vol. 45, no.5, pp. 940-943, 2000

2000
[7]

Trinh and T

H. Trinh and T. Fernando, Functional Observers for Dynamical Systems . Springer-V erlag, Berlin Heidelberg, 2012. 5

2012

[1] [1]

Yu, Critical Ledgers and Scale-Defect Cascades for Navier–Stokes, arXiv preprint arXiv:2606.13887 [math.AP], 2026

H. Trinh, “Delayed functional observers for output-del ayed linear sys- tems”, Preprint at https://doi.org/10.48550/arXiv.2606 .09407 (2026)

work page doi:10.48550/arxiv.2606 2026

[2] [2]

On Time-Delay Compensators for Delayed-Output Systems

H. Trinh, “On time-delay compensators for delayed-outp ut systems”, Preprint at https://doi.org/10.48550/arXiv.2606.10308 (2026)

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

[3] [3]

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 internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.03081 2026

[4] [4]

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 internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.17434 2026

[5] [5]

Existence and design of functional observers for time-delay systems with delayed output measu rements

H. Trinh, P . T. Nam and T. Fernando, “Existence and design of functional observers for time-delay systems with delayed output measu rements”, Preprint at https://doi.org/10.48550/arXiv.2603.09395 (2026)

work page doi:10.48550/arxiv.2603.09395 2026

[6] [6]

Existence and design of functional observ ers

M. Darouach, “Existence and design of functional observ ers”, IEEE Trans. Autom. Contr ., vol. 45, no.5, pp. 940-943, 2000

2000

[7] [7]

Trinh and T

H. Trinh and T. Fernando, Functional Observers for Dynamical Systems . Springer-V erlag, Berlin Heidelberg, 2012. 5

2012