On Time-Delay Compensators for Delayed-Output Systems

Hieu Trinh

arxiv: 2606.10308 · v1 · pith:LOBCK2LDnew · submitted 2026-06-09 · 📡 eess.SY · cs.SY

On Time-Delay Compensators for Delayed-Output Systems

Hieu Trinh This is my paper

Pith reviewed 2026-06-27 12:43 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords functional observerstime-delay systemsdelayed outputsobserver designlinear systemsstate estimationsensing latency

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The pith

Incorporating an extra delayed measurement into a multi-delay observer allows functional estimation when output delay is shorter than state delay.

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

The paper establishes that a functional observer structure using multiple internal delays and an augmented architecture can estimate z(t) = F x(t) in linear time-delay systems even when the independent measurement delay h satisfies 0 < h < τ. It compensates for sensing latency by deriving algebraic existence conditions and a constructive synthesis procedure. A sympathetic reader would care because the added delayed measurement vector widens the range of systems that admit such observers, including those with larger overall delays. The approach focuses on constant-delay linear systems and shows how the extra measurement relaxes prior restrictions on delay sizes.

Core claim

The central claim is that the proposed functional observer with multiple internal delays and an augmented architecture, by incorporating an additional delayed measurement vector, satisfies algebraic existence conditions for estimating the functional z(t) = F x(t) in linear systems where the measurement delay h is independent of and smaller than the state delay τ, thereby applying to a wider class of systems with larger delays.

What carries the argument

The multi-delay functional observer structure with augmented architecture and additional delayed measurement vector, which generates the required estimates through internal delay chains and compensates for the mismatch between h and τ.

If this is right

Algebraic existence conditions are provided that determine when the observer can be designed for a given system.
A constructive synthesis procedure yields the observer parameters once the conditions hold.
The design applies to a wider class of systems than before, including those with larger state and output delays.
The approach expands the set of admissible delay combinations for which functional estimation is possible.

Where Pith is reading between the lines

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

The structure suggests that similar compensation via extra delayed channels could be tested in discrete-time or sampled-data versions of the same systems.
The algebraic conditions might serve as a starting point for checking feasibility when delays are approximately constant but slowly varying.
Closed-loop use of the observer could be examined to see whether the expanded design space translates into stabilizable plants with larger delays.

Load-bearing premise

The system is linear with constant delays satisfying 0 < h < τ, and the algebraic existence conditions derived from the proposed multi-delay observer structure can be satisfied for the given system matrices.

What would settle it

A specific linear system with constant delays 0 < h < τ for which no observer gains exist that satisfy the algebraic conditions even after adding the extra delayed measurement vector, or where the resulting observer produces a nonzero steady-state estimation error for z(t).

read the original abstract

This paper advances the practical utility of functional observer theory by addressing sensing latency in linear time-delay systems. We address the estimation of the functional $z(t)=Fx(t)$ in cases where the measurement delay $h$ is independent of the internal state delay $\tau$, with a specific focus on the condition $0 < h < \tau$. To compensate for sensing lags, we propose a functional observer structure characterized by multiple internal delays and an augmented architecture. Algebraic existence conditions are established alongside a constructive synthesis procedure. By incorporating an additional delayed measurement vector, we demonstrate that this approach significantly expands the design space and is applicable to a wider class of systems with larger state and output 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

The paper fills a gap for functional observers in linear time-delay systems with 0 < h < τ using a multi-delay structure and extra measurement.

read the letter

The punchline is that this work fills a gap in functional observer design for linear time-delay systems by tackling the case where the output delay h is positive but less than the state delay τ. They do this with a multi-delay observer structure that includes an additional delayed measurement vector.

What the paper does is establish algebraic existence conditions for this setup and provide a synthesis procedure to build the observer. This is new because previous approaches apparently didn't cover this independent delay mismatch directly. The extra measurement expands the possibilities, allowing application to systems with bigger delays overall.

It handles the problem by augmenting the observer to compensate for the sensing lag. The derivation seems straightforward from the structure, and there's no sign of circular reasoning or self-referential fitting.

One soft spot is the lack of concrete examples or numerical validation in the provided abstract. It would be good to see if the conditions are met for typical systems or if there are cases where it fails despite the claims. Also, the paper sticks to linear constant delay systems, so it doesn't address time-varying or nonlinear cases, but that's not a flaw for what it sets out to do.

This paper is for researchers in systems and control theory, particularly those working on observers for delayed systems in applications like process control or communication networks. A reader looking for extensions in functional observer theory would get value from the constructive method.

The work shows clear thinking on the literature gap and honest engagement with the specific problem. It deserves a serious referee because the contribution is focused and the method is algebraic and constructive. I would recommend sending it for peer review to verify the derivations and assess the practical impact.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes a functional observer for linear time-delay systems to estimate z(t)=Fx(t) when the output measurement delay h is independent of the state delay τ (specifically 0<h<τ). It introduces an augmented observer architecture with multiple internal delays that incorporates an additional delayed measurement vector, derives algebraic existence conditions for this structure, and supplies a constructive synthesis procedure. The central claim is that this multi-delay approach expands the design space relative to prior methods and applies to a wider class of systems with larger delays.

Significance. If the algebraic conditions and synthesis procedure are valid, the work offers a concrete advance in functional observer design for time-delay systems by relaxing restrictions on the relative sizes of state and output delays. The constructive procedure is a positive feature that could facilitate practical implementation in applications such as networked control.

major comments (1)

[Abstract] Abstract: the assertion that the method 'significantly expands the design space' and applies to 'a wider class of systems with larger state and output delays' is load-bearing for the contribution, yet the provided text supplies neither the explicit algebraic conditions nor any comparative example or counter-example showing a system where prior observers fail but the new structure succeeds.

minor comments (1)

The notation for the functional matrix F, delays h and τ, and the augmented observer matrices should be introduced with explicit definitions before the existence conditions are stated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment on the abstract. We address the concern point by point below and indicate where revisions will be made to strengthen the presentation of the contribution.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that the method 'significantly expands the design space' and applies to 'a wider class of systems with larger state and output delays' is load-bearing for the contribution, yet the provided text supplies neither the explicit algebraic conditions nor any comparative example or counter-example showing a system where prior observers fail but the new structure succeeds.

Authors: We agree that the abstract, being a concise summary, does not embed the full algebraic conditions or a detailed counter-example. These elements are developed in the body of the manuscript: the algebraic existence conditions (rank conditions on the system matrices together with the delay-dependent solvability equations) are stated explicitly, and the constructive synthesis procedure that incorporates the additional delayed measurement is given in detail. The claim of an expanded design space is supported by the fact that the new structure applies precisely when 0 < h < τ, a regime excluded by prior single-delay or matched-delay observers. Numerical illustrations in the manuscript demonstrate successful estimation on systems satisfying this independent-delay condition. To make the abstract self-contained with respect to the load-bearing claim, we will revise it to briefly reference the key delay regime 0 < h < τ and note that the examples confirm applicability beyond existing methods. This is a partial revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper proposes a novel multi-delay functional observer architecture for linear systems with independent measurement delay h and state delay τ (0 < h < τ), then directly derives algebraic existence conditions and a constructive synthesis procedure from the augmented observer equations. No load-bearing step reduces by definition to its own inputs, renames a fitted parameter as a prediction, or relies on a self-citation chain whose validity is presupposed; the central claims are self-contained constructive results for an expanded design space.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The contribution rests on standard domain assumptions of linear time-delay systems theory; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

domain assumption The plant is a linear system with constant state delay τ and constant output delay h.
Invoked throughout the abstract as the setting for the observer design.
domain assumption The functional to be estimated is of the form z(t) = F x(t) with known matrix F.
Core premise of functional observer theory stated in the abstract.

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discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Delayed Functional Observers for the Realization of Generalized Delayed Control Laws
eess.SY 2026-06 unverdicted novelty 2.0

Design of delayed functional observers for estimating generalized delayed control laws to stabilize time-delay systems with mismatched input and output delays.

Reference graph

Works this paper leans on

4 extracted references · 2 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

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 measurements”, Preprint at https://doi.org/10.48550/arXiv.2603.09395 (2026)

work page doi:10.48550/arxiv.2603.09395 2026
[2]

Time-Delay Compensators for Linear Systems with Delayed Output Measurements

H. Trinh, P. T. Nam and T. N. Nguyen, “Time-delay compensators for linear systems with delayed output measurements”, Preprint 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]

Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method,

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

2012
[4]

Criteria for asymptotic stability of linear time-delay systems,

T. Mori, “Criteria for asymptotic stability of linear time-delay systems,” IEEE Transactions on Automatic Control, vol. 30, no. 2, pp. 158-161, 1985. 6

1985

[1] [1]

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 measurements”, Preprint at https://doi.org/10.48550/arXiv.2603.09395 (2026)

work page doi:10.48550/arxiv.2603.09395 2026

[2] [2]

Time-Delay Compensators for Linear Systems with Delayed Output Measurements

H. Trinh, P. T. Nam and T. N. Nguyen, “Time-delay compensators for linear systems with delayed output measurements”, Preprint 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]

Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method,

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

2012

[4] [4]

Criteria for asymptotic stability of linear time-delay systems,

T. Mori, “Criteria for asymptotic stability of linear time-delay systems,” IEEE Transactions on Automatic Control, vol. 30, no. 2, pp. 158-161, 1985. 6

1985