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arxiv: 2604.17434 · v1 · submitted 2026-04-19 · 📡 eess.SY · cs.SY

Time-Delay Compensators for Linear Systems with Delayed Output Measurements

H. Trinh , P. T. Nam , T. N. Nguyen This is my paper

Pith reviewed 2026-05-10 05:48 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords functional observersdelayed output measurementstime-delay compensationlinear systemsasymptotic stabilityobserver designstate estimationerror dynamics
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The pith

Linear systems with delayed sensor data can still yield accurate current-state estimates via observers built from multiple delayed measurement copies.

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

The paper develops a design method for functional observers that reconstruct a linear combination of the state at the present time, using only measurements that arrive after a fixed delay. Rather than estimating the delayed state and then predicting forward, the observer directly generates the current functional by incorporating several time-shifted versions of the output into its structure. This architecture relaxes the usual restrictions on how large the delay may be while still guaranteeing that the estimation error tends to zero. Explicit conditions for the existence of such observers are derived, together with algebraic procedures to compute the required gains. The result enlarges the set of practical situations in which functional observers remain usable when outputs cannot be sampled instantaneously.

Core claim

For a linear system with output delayed by h, a functional observer of the form that combines multiple delayed output terms and internal dynamics can be synthesized so that the error between the estimated functional and the true z(t) = F x(t) converges asymptotically to zero, provided matrix inequalities or rank conditions involving the system matrices and the chosen delays are satisfied; this construction permits strictly larger values of h than single-delay observers allow.

What carries the argument

Multi-delayed functional observer that augments the standard Luenberger structure with several parallel delayed copies of the output to cancel latency in the error equation.

If this is right

  • The allowable sensor delay can be increased without destroying asymptotic stability of the estimation error.
  • Observer synthesis reduces to solving a set of linear matrix inequalities or rank conditions once the delayed components are introduced.
  • The same framework recovers standard Luenberger observers as a special case when only one delay is used.
  • Numerical checks confirm that the constructed observers remain stable for delay values beyond the limits of conventional designs.

Where Pith is reading between the lines

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

  • The structure may be tested on systems whose delays arise from communication networks rather than pure sensors.
  • If the delays can be chosen freely during design, optimization routines could maximize the stability margin for a given plant.
  • Extension to observers that also reject disturbances or track references would follow by adding feedforward terms to the same multi-delay skeleton.

Load-bearing premise

The underlying system is linear and there exist observer parameters, including the number and values of the delays, that make the resulting error-system matrix Hurwitz.

What would settle it

A concrete linear system and delay h for which the paper's existence conditions hold, yet every choice of the multiple delay coefficients and observer matrices leaves at least one eigenvalue of the error matrix with positive real part.

Figures

Figures reproduced from arXiv: 2604.17434 by H. Trinh, P. T. Nam, T. N. Nguyen.

Figure 1
Figure 1. Figure 1: Schematic diagram of an observer acting as a time-del [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: shows trajectories of x1(t), xˆ1(t) and y1(t) = x1(t− 1). It is clear that asymptotic estimation has been achieved [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Trajectories of x2(t), xˆ2(t) and y2(t) = x2(t − 1) output measurement contains only one delayed state variable x1(t − τ), i.e., y(t) = 1 0 x(t − τ). Again, our objective is to design an observer to estimate instantaneously the entire state vector x(t) using the above delayed output measurement. With Cτ = [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schematic diagram of an observer-based feedback con [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Trajectories of x1(t) using functional observer-based control and state-feedback control, and reference signal r(t) VII. CONCLUSION This paper has established a comprehensive framework for designing functional observers for linear systems subject to output measurement delays. Moving beyond traditional methodologies, the proposed observer generates an estimate zˆ(t) that predicts the current state functiona… view at source ↗
read the original abstract

This paper provides a comprehensive framework for designing functional observers for linear systems subject to delayed output measurements. Moving beyond traditional methodologies, the proposed observer generates an estimate $\hat{z}(t)$ that predicts the current state functional $z(t)=Fx(t)$ using delayed data. By neutralizing sensing latency, the observer serves as a potent time-delay compensator, effectively expanding the practical utility of functional observer theory. The proposed observer architecture offers greater robustness and versatility than traditional Luenberger-type observers by leveraging multiple delayed components to preserve accuracy despite latency. A key contribution of this work is a novel method for extending the maximum allowable measurement delay while maintaining the asymptotic stability of the estimation-error system. Existence conditions are established together with constructive synthesis procedures. Extensive numerical examples are given to illustrate the proposed theory.

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.

Referee Report

0 major / 4 minor

Summary. The manuscript develops a framework for functional observers for linear time-invariant systems with delayed output measurements. The proposed observer architecture employs multiple delayed output components to generate an estimate of the current functional z(t) = F x(t), thereby compensating for the measurement delay. Existence conditions (presumably rank or LMI-based) are derived for the observer parameters, together with constructive synthesis procedures, such that the estimation-error system remains asymptotically stable. A central claim is that this architecture extends the maximum allowable delay beyond what is possible with conventional single-delay or Luenberger-type observers. The results are illustrated with numerical examples.

Significance. If the derivations hold, the work is significant for practical estimation and control in systems where sensor latency is unavoidable. The multiple-delay-component architecture provides a concrete way to enlarge the feasible delay interval while preserving stability, which is a useful extension of functional-observer theory. Credit is due for the explicit existence conditions, the synthesis procedure, and the numerical validation that demonstrates the delay-extension property. The linear-system setting and standard stability analysis are compatible with the claimed architecture.

minor comments (4)
  1. [Abstract] Abstract: the phrase 'neutralizing sensing latency' is informal; replace with a precise statement about the error dynamics.
  2. [§3] §3 (existence conditions): the rank or LMI conditions should be stated explicitly with the precise matrix dimensions and the role of each delayed component made clear.
  3. [Numerical examples] Numerical examples: a table or plot directly comparing the maximum allowable delay achieved by the proposed observer versus a standard functional observer would strengthen the central claim of extension.
  4. [§2] Notation: the distinction between the different delay values in the observer equations should be introduced earlier and used consistently throughout.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our manuscript on functional observers for systems with delayed output measurements. We appreciate the recommendation for minor revision and the recognition of the framework's potential to extend allowable delays while preserving stability.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper derives a functional observer architecture for systems with delayed outputs by constructing an error dynamics equation, then establishes existence conditions (rank or LMI-type) and constructive synthesis procedures that guarantee asymptotic stability for a range of delays. These steps rely on standard linear systems analysis and do not reduce the stability claim or the extended delay bound to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. Numerical examples illustrate the conditions but are not used to force the central result. The derivation remains self-contained against external benchmarks such as Lyapunov stability criteria.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no specific free parameters, axioms, or invented entities can be identified; the work builds on standard linear systems and observer theory without introducing new entities or ad-hoc assumptions visible here.

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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    2 0 ) , and τ = 1. 2. For τ = 1 . 2, Remark A2 ensures that system (87) is asymptotically stable for a maximum allowable time delay h = 1. 68. B. Stabilization of time-delay systems In this section, based on the stability conditions derived in Section VIII-A, we investigate the stabilization proble m for systems with one or multiple time delays. First, we...

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    2 0 . 1 − 0. 1 0 0 . 2 0 . 15   . Noting that, in this example, the matrix N is unstable. For Case 1, by applying Lemma 9, we obtain a maximum allowable delay τ = 4. 8 and the matrix Nτ =   − 0. 2033 0 . 0001 − 0. 0004 − 0. 1619 − 0. 1403 0 . 0839

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    1707 − 0

    0195 − 0. 1707 − 0. 1751   , which guarantees that the closed-loop system is asymptotic ally stable. For Case 2, by applying Lemma 10, we obtain an allowable delay interval τ ∈ [2, 4. 78] and the matrix Nτ =   − 0. 2066 0 . 0004 0 . 0003 − 0. 1570 − 0. 1460 0 . 0833

    work page 2066

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    1751 − 0

    0174 − 0. 1751 − 0. 1731   , which ensures that the closed-loop system is asymptoticall y stable for any time-delay τ ∈ [2, 4. 78]. 17 Example A5: Consider Problem 2 for system (47) with the matrices N0, 1 =  

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    2 0 . 1 − 0. 1 0 0 . 2 0 . 15   , N 0, 2 = ( 0 0 0 ) , Nτ, 1 = 03× 3, N τ, 2 = ( 1 2 3 ) . We aim to determine a matrix Z ∈ R3× 1 such that the closed- loop system ˙e(t) = ( N0, 1 + ZN 0, 2)e(t) + (Nτ, 1 + ZN τ, 2)e(t − τ), (113) is asymptotically stable. For Case 1, by applying Lemma 11, we obtain a maximum allowable delay τ = 2. 2 and the matrix Z = ...

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    2 0 . 1 − 0. 1 0 0 . 2 0 . 15   , N 0, 2 = ( 0 0 0 ) , Nτ, 1 = Nh, 1 = 03× 3, N τ, 2 = Nh, 2 = ( 1 2 3 ) , and 0 < τ < h . We aim to determine matrices Z0, Z τ , Z h ∈ R3× 1, the largest possible time delay τ, and a time delay h > τ such that the closed-loop system (87) is asymptotically sta ble. By Lemma 13, we obtain τ = 2. 43, h = 2. 7, and Z0 =   ...

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  16. [16]

    Notably, in the case where Nh, 2 = 01× 3, i.e., when stabilizing system (87) using only a delayed feedback term, Lemma 11 yields a smaller allowable time delay τ = 2

    0524   , which ensure that the closed-loop system (87) is asymptoti- cally stable. Notably, in the case where Nh, 2 = 01× 3, i.e., when stabilizing system (87) using only a delayed feedback term, Lemma 11 yields a smaller allowable time delay τ = 2 . 2 (see Example A5). Again, this demonstrates that employing two delayed feedback terms can increase the ...

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  17. [17]

    2 0 . 1 − 0. 1 0 0 . 2 0 . 15   , N 1, 1 = N2, 1 = N3, 1 = 03× 3, N0, 2 = ( 0 0 0 ) , N 1, 2 = N2, 2 = ( 1 2 3 ) , N3, 2 = ( 1 0 0 ) . and τ1 = 3. 65, τ 2 = 3. 7, τ 3 = 3. 75. We aim to determine matrices Z0, Z 1, Z 2, Z 3 ∈ R3× 1, such that the closed-loop system (75) is asymptotically stable. By Lemma 14, we obtain Z0 =   0 0 0   , Z 1 =   − 0. ...

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  18. [18]

    2249 − 0

    9271   , Z 3 =   − 0. 2249 − 0. 1297

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  19. [19]

    which ensure that the closed-loop system (75) is asymptoti- cally stable

    0445   . which ensure that the closed-loop system (75) is asymptoti- cally stable. Compared with Example A7, where system (75) is stabilized with τ = 2 . 43, the present example shows that system (75) remains stable for a larger delay of τ1 = 3 . 65. This indicates that, compared to using only two delayed feedback terms, employing three delayed feedback...

    work page