Delayed Functional Observers for Output-Delayed Linear Systems

Hieu Trinh

arxiv: 2606.09407 · v1 · pith:FMT5RKLDnew · submitted 2026-06-08 · 📡 eess.SY · cs.SY

Delayed Functional Observers for Output-Delayed Linear Systems

Hieu Trinh This is my paper

Pith reviewed 2026-06-27 15:30 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords delayed functional observersoutput-delayed linear systemsactuator and sensor delayscontrol law reconstructiondual-channel latencylow-order observersnetworked control systems

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The pith

Delayed functional observers reconstruct control laws under simultaneous unequal actuator and sensor delays without full-state estimation.

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

This paper introduces a class of delayed functional observers for linear systems that experience output measurement lags. The approach targets the reconstruction of delayed control laws when actuator and sensor channels carry different delays at the same time. A sympathetic reader would care because the design stays low-order and skips both complete state reconstruction and heavy distributed computation. If the method works, it directly supports control implementation in networked setups where delays cannot be removed.

Core claim

The paper claims that a novel class of delayed functional observers can systematically mitigate simultaneous, unequal delays across both the actuator and sensor channels in linear systems, enabling reconstruction of delayed control laws without requiring full-state estimation or computationally intensive real-time distributed integration.

What carries the argument

Delayed functional observers that compensate dual-channel latency in output-delayed linear systems.

If this is right

Dual-channel latency is resolved without estimating the full system state.
Computation stays low because real-time distributed integration is not needed.
Control laws can be reconstructed directly from delayed outputs.
The framework applies to practical networked engineering systems with output lags.

Where Pith is reading between the lines

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

The same observer structure might be tested on systems where one delay is fixed and the other varies slowly.
Integration with existing delay-compensating controllers could reduce overall order further.
Numerical checks on benchmark linear systems with known delays would confirm reconstruction error bounds.

Load-bearing premise

The linear system must admit a low-order functional observer design that simultaneously compensates the unequal actuator and sensor delays.

What would settle it

A concrete linear system with specified unequal delays for which no low-order observer exists that reconstructs the required control law accurately.

Figures

Figures reproduced from arXiv: 2606.09407 by Hieu Trinh.

**Figure 1.** Figure 1: 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) -100 -50 0 50 x 1 (t) x 2 (t) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Trajectories of e1(t) Furthermore, as noted in Example 1, the LMI condition in Lemma 11 of [2] remains feasible for delays up to the maximum upper bound of τM = 1.26s. In contrast, for τ˜ = 1.48s and τ˜ + α = 2s, the LMI condition in Lemma 13 of [2] remains feasible, thereby ensuring the asymptotic stability of (14). Example 3: This example is provided to further illustrate the practical implications of Re… view at source ↗

**Figure 3.** Figure 3: Trajectories of x1(t) and x2(t) III. CONCLUSION In this paper, a novel class of delayed functional observers has been successfully developed to reconstruct delayed control laws subject to severe output measurement lags, thereby extending the foundational results reported in [1], [2]. The core strength of the proposed architecture lies in its ability to systematically mitigate simultaneous, unequal delays … view at source ↗

read the original abstract

This paper introduces a novel class of delayed functional observers specifically designed to reconstruct delayed control laws under severe output measurement lags, directly complementing recent literature \cite{trinhnn26, trinhnam26}. By systematically mitigating simultaneous, unequal delays across both the actuator and sensor channels, the proposed architecture resolves dual-channel latency without requiring full-state estimation or computationally intensive real-time distributed integration. Ultimately, this work provides a powerful, low-order framework that bridges the gap between idealized control theory and the practical constraints of modern networked engineering systems.

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

Incremental extension of the author's prior functional observer work to dual unequal delays, but no existence conditions are supplied so generality is unclear.

read the letter

This paper extends functional observer designs to linear systems where actuator and sensor channels have different constant delays. The architecture aims to reconstruct the delayed control input directly with a low-order observer, avoiding full-state reconstruction and distributed integration.

The practical angle is the strongest part. Networked control often has mismatched delays, and skipping full-state estimation plus heavy computation is a reasonable target for that setting.

The soft spot is the lack of any stated conditions. The abstract gives no rank requirements on the system matrices, no observability assumptions, and no bounds on the delay sizes that would guarantee the observer exists. If the full paper only assumes the system admits such a design without explicit algebraic or LMI tests, the result applies only to an unspecified subset of systems. The framing as a direct complement to the author's own recent papers also keeps the step forward small.

This is for specialists already working on delay compensation and functional observers in linear systems. A reader in that narrow area might pick up the architecture if the derivations are complete.

I would send it for peer review so the technical claims can be checked against the actual conditions and proofs.

Referee Report

2 major / 1 minor

Summary. The paper proposes a novel class of delayed functional observers for linear systems (A,B,C) with known but unequal constant delays au_a (actuator) and au_s (sensor). The observers are claimed to reconstruct the delayed control law at low order, without full-state estimation or real-time distributed integration, thereby addressing dual-channel latency in networked control.

Significance. If explicit existence conditions and constructive design procedures are supplied and verified, the result could offer a computationally lighter alternative to full-order observers or distributed-delay compensators in output-delayed systems, with potential applicability to networked engineering applications.

major comments (2)

[Abstract, §1] Abstract and §1: The central claim that the architecture 'resolves dual-channel latency' for arbitrary linear systems rests on the unstated assumption that (A,B,C) admits a functional observer of order strictly less than n that simultaneously compensates unequal delays; no rank, observability, or delay-bound conditions (e.g., LMI feasibility tests) are supplied, so the domain of applicability remains unspecified.
[Abstract] The manuscript does not derive or state algebraic or LMI conditions guaranteeing the existence of the low-order delayed functional observer; without these, the architecture applies only to an unspecified subset of systems and the generality asserted in the abstract cannot be assessed.

minor comments (1)

[Abstract] The abstract's phrasing 'directly complementing recent literature [trinhnn26, trinhnam26]' should be expanded in the introduction to delineate the precise technical advance relative to those works.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments highlight the need for greater clarity regarding existence conditions, which we address below with corresponding revisions to strengthen the paper.

read point-by-point responses

Referee: [Abstract, §1] Abstract and §1: The central claim that the architecture 'resolves dual-channel latency' for arbitrary linear systems rests on the unstated assumption that (A,B,C) admits a functional observer of order strictly less than n that simultaneously compensates unequal delays; no rank, observability, or delay-bound conditions (e.g., LMI feasibility tests) are supplied, so the domain of applicability remains unspecified.

Authors: We agree that the manuscript would benefit from explicit statements of the conditions under which the proposed low-order delayed functional observer exists. In the revision, we will add a dedicated subsection (likely in Section 3) that derives the necessary rank and observability conditions on the triple (A, B, C) together with delay-dependent bounds that guarantee an observer order strictly less than n. We will also include LMI feasibility tests for practical verification. These additions will precisely delineate the domain of applicability without altering the core contribution. revision: yes
Referee: [Abstract] The manuscript does not derive or state algebraic or LMI conditions guaranteeing the existence of the low-order delayed functional observer; without these, the architecture applies only to an unspecified subset of systems and the generality asserted in the abstract cannot be assessed.

Authors: The original presentation emphasizes the constructive design procedure once the observer parameters are assumed to exist, but we accept that explicit algebraic and LMI conditions are required to support the generality claims. We will revise the abstract to qualify the claims and insert the missing conditions (rank conditions on augmented delay-compensated matrices plus LMI solvability) into the main body. This will allow readers to determine applicability for any given (A, B, C, τ_a, τ_s). revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation presented as novel without reduction to self-citations or inputs

full rationale

The abstract references prior literature by overlapping authors only to position the work as complementary, without invoking any uniqueness theorem, ansatz, or fitted parameter from those citations to justify the central observer architecture. No equations, existence conditions, or derivation steps are supplied in the provided text that reduce the claimed low-order delayed functional observer to its own inputs by construction. The paper states a novel class resolving dual-channel latency without full-state estimation, and absent any exhibited self-definitional loop or fitted-input prediction, the chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no equations or design details; therefore no free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.1-grok · 5601 in / 994 out tokens · 22623 ms · 2026-06-27T15:30:38.489998+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages · 2 internal anchors

[1]

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
[2]

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
[3]

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). 5

work page doi:10.48550/arxiv.2603.09395 2026

[1] [1]

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

[2] [2]

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

[3] [3]

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). 5

work page doi:10.48550/arxiv.2603.09395 2026