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arxiv: 2606.03081 · v1 · pith:7XKMFXFMnew · submitted 2026-06-02 · 📡 eess.SY · cs.SY

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

Hieu Trinh , Phan Thanh Nam , Tran Ngoc Nguyen This is my paper

Pith reviewed 2026-06-28 09:21 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords linear systemstime delaysmismatched delaysobserver-based controlLMIstate-feedbackoutput feedback
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The pith

Linear systems with independent input and output delays are stabilized by realizing an LMI-designed delayed state-feedback controller through time-delay compensators.

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

The paper shows that linear systems subject to mismatched delays in the input and output can be stabilized without forcing the delays to match. It first obtains an asymptotically stabilizing delayed state-feedback law using existing LMI methods, then implements that law with compensators that separately handle the output measurement delay. A sympathetic reader would care because mismatched delays appear in many practical settings such as networked or remote control, and the method lets the designer keep the original controller design while adding only the compensators. The same architecture is shown to work when only output measurements are available.

Core claim

The paper claims that an asymptotically stabilizing delayed state-feedback controller obtained via LMI techniques can be realized by novel time-delay compensators, thereby accommodating an output measurement delay that is independent of the input delay and enabling direct estimation of the control law; the construction extends directly to target output controllers under the same mismatched-delay conditions.

What carries the argument

The pairing of an LMI-synthesized delayed state-feedback controller with time-delay compensators that separately reconstruct the effect of the output delay.

If this is right

  • The closed-loop system remains asymptotically stable under the combined controller and compensators.
  • The output delay can differ arbitrarily from the input delay while the control law is still estimated directly.
  • The same compensator structure works when only output measurements are available, yielding a target output controller.

Where Pith is reading between the lines

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

  • The compensators could be attached to any existing LMI-designed delayed controller without requiring a fresh stability proof.
  • The separation of input and output delay handling might extend to cases where one delay is time-varying if the underlying LMI conditions admit suitable adaptation.

Load-bearing premise

A stabilizing delayed state-feedback controller for the system exists and can be found by LMI methods, and the compensators can be added to it without creating new instability.

What would settle it

A concrete linear system for which an LMI yields a stabilizing delayed state-feedback law, yet closed-loop simulation with the compensators under mismatched delays shows divergence.

Figures

Figures reproduced from arXiv: 2606.03081 by Hieu Trinh, Phan Thanh Nam, Tran Ngoc Nguyen.

Figure 1
Figure 1. Figure 1: Trajectories of x1(t) and x2(t) delayed functional z(t) = F x(t − τu) using the delayed output vector y(t) = Cx(t − τy). Building on established techniques for instantaneous functional estimation [1], [6], our method focuses on the direct reconstruction of the delayed state functional. Let us look at the following observer which incorporates an internal delay dynamics and it is driven by the delayed output… view at source ↗
Figure 2
Figure 2. Figure 2: Trajectories of x1(t) and x2(t) Remark 4: While the LMI condition in Lemma 11 [1] is technically feasible for output delays up to τ¯y, the observer structure in (11)-(12) fails to remain effective if τ¯y ≤ τ¯u. To address this, the permissible output delay τy must be expanded. Following the approach in [1], this is accomplished by using an augmented measurement vector within a generalized observer framewor… view at source ↗
Figure 3
Figure 3. Figure 3: Trajectories of e1(t) and e2(t) ˙zˆaug(t) =  1.0429 0.8484 −1.2314 −0.0429 zˆaug(t) +  −0.8705 −0.3991 0.9424 −0.4705 zˆaug(t − 0.75) +  −0.0650 −0.0002 −0.1714 −0.0400 zˆaug(t − 1.45) +  0.7070 −0.3382 −0.7654 1.2687  y(t − 0.05) +  0.0528 −0.0548 0.1392 −0.1051 y(t − 0.75) +  −0.8469 1  u(t − 1.5). To illustrate the validity of the observer-based control scheme which combines the controller a… view at source ↗
Figure 4
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) -15 -10 -5 0 5 10 15 20 25 x 1 (t) x 2 (t) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Trajectories of zo(t) and za(t) under the proposed target output controller output controllable [2]. Since target output controllability is a more relaxed condition than the controllability condition. This benefit previously reported in [2]. Next, we consider the practical scenarios where neither zo(t) nor its delayed counterpart zo(t−τu) is directly available for feedback. Additionally, the measured outpu… view at source ↗
Figure 6
Figure 6. Figure 6: Trajectories of zo(t) and za(t) under the proposed observer-based delayed target output controller D. Design of Delayed Functional Observers via Row-Space State Projection Section II-C presented the design of delayed functional observers for estimating the target output controllers (30)-(31) using the full-order system (1)-(2). Alternatively, consider the reduced-order subsystem (27), constructed by projec… view at source ↗
Figure 7
Figure 7. Figure 7: Trajectories of zo(t) under the proposed target output controller Next, based on the derived second-order system, we design a functional observer to estimate the control input functional defined as z(t) := u(t − 0.5) = Z˜ τu z˜o(t − 0.5), where the observer utilizes only one delayed output measure￾ment yo(t) = y1(t) = 0 1 z˜o(t − 0.7). As discussed in Scenario 2 of Section II-B, we employ ob￾server (11)-(… view at source ↗
Figure 8
Figure 8. Figure 8: Trajectories of zo(t) under the proposed observer-based target output controller Example 5: We adopt Example 4 from [2] to design an observer-based target controller for the system where it is subject to mismatched time delays in both the state and output vectors, where τu = 1.7s and τy = 3s. The system matrices are given by A =   −0.5 0.5 −1 −0.5 0.5 −0.7 −0.5 1.4 0.7 −0.7 −0.6 0 0.2 0.6 −0.6 0.25 0… view at source ↗
Figure 9
Figure 9. Figure 9: Trajectories of vector zo(t) under the proposed target output controller Although Fo = C, the heavy measurement delay (τy > τu) prevents the direct implementation of the designed control law. This limitation arises because the required state vector zo(t−1.7) is unavailable, and only zo(t−3) can be accessed. To overcome this, we can leverage the framework developed in Scenario 2 (Section II-B) and utilize e… view at source ↗
Figure 10
Figure 10. Figure 10: Trajectories of zo(t) under the proposed observer-based target output controller III. CONCLUSION This paper investigated the stabilization of linear systems subject to simultaneous, mismatched input and output time delays. First, an asymptotically stabilizing delayed state￾feedback controller was synthesized using advanced LMI techniques. Second, this controller was realized via novel time￾delay compensat… view at source ↗
read the original abstract

This paper investigates the stabilization of linear systems subject to simultaneous, mismatched time delays in both the control input and system output vectors. The proposed control framework is developed in two primary stages. First, an asymptotically stabilizing delayed state-feedback controller is synthesized by leveraging recent advancements in Linear Matrix Inequality (LMI) techniques. Second, this controller is realized using novel time-delay compensators \cite{trinhnam26}. This architecture successfully accommodates an output measurement delay $\tau_y$ that is independent of the input delay $\tau_u$, enabling direct estimation of the delayed state-feedback control law. The proposed methodology is then extended to target output controllers to account for simultaneous, mismatched time delays in both the control input and system output vectors.

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

3 major / 2 minor

Summary. The paper proposes a two-stage observer-based control architecture for linear systems subject to independent (mismatched) input delay τ_u and output measurement delay τ_y. Stage 1 synthesizes an asymptotically stabilizing delayed state-feedback law via existing LMI techniques; Stage 2 realizes that law using novel time-delay compensators from the cited reference trinhnam26. The architecture is claimed to accommodate independent τ_y without requiring τ_y = τ_u, and the method is extended to output-feedback controllers.

Significance. If a rigorous composite stability certificate were supplied, the result would furnish a systematic LMI-based route to output-feedback stabilization under independent input/output delays, extending recent delay-compensation techniques. The absence of any LMI formulation, closed-loop Lyapunov or LMI analysis, or numerical example in the manuscript, however, leaves the practical significance currently unsupported.

major comments (3)
  1. [Abstract / methodology overview] Abstract and methodology description: the central claim that the LMI-synthesized delayed state-feedback can be realized by the compensators of trinhnam26 while an observer handles independent τ_y rests on the unproven assertion that no additional stability conditions arise from the observer error dynamics. No composite Lyapunov function or augmented LMI is exhibited to certify the interconnection.
  2. [Stage-1 controller synthesis] The manuscript invokes 'recent advancements in LMI techniques' for the delayed state-feedback design yet supplies neither the explicit LMI conditions nor the plant model (including how the input delay enters the state equation) used to obtain the controller.
  3. [Output-feedback extension] Extension to output controllers: the claim that the same compensator-plus-observer architecture works for dynamic output feedback likewise lacks any stability argument or LMI that accounts for the simultaneous presence of both delays in the closed-loop map.
minor comments (2)
  1. Notation for the two delays (τ_u, τ_y) and the compensator blocks should be introduced with a single diagram or block diagram early in the paper.
  2. The citation trinhnam26 appears only in the abstract; its precise statement (assumptions on the plant, matched vs. mismatched delays) should be recalled in the main text before the combination is asserted.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and indicate the revisions that will be made to strengthen the stability arguments and explicit formulations.

read point-by-point responses

  1. Referee: [Abstract / methodology overview] Abstract and methodology description: the central claim that the LMI-synthesized delayed state-feedback can be realized by the compensators of trinhnam26 while an observer handles independent τ_y rests on the unproven assertion that no additional stability conditions arise from the observer error dynamics. No composite Lyapunov function or augmented LMI is exhibited to certify the interconnection.

    Authors: We agree that an explicit composite stability certificate is required. The manuscript currently relies on the realization properties established in the cited compensator reference without deriving a joint Lyapunov-Krasovskii functional for the observer-compensator interconnection. In the revision we will add a dedicated stability section that constructs such a functional and derives the associated LMI conditions guaranteeing asymptotic stability of the overall closed-loop system. revision: yes

  2. Referee: [Stage-1 controller synthesis] The manuscript invokes 'recent advancements in LMI techniques' for the delayed state-feedback design yet supplies neither the explicit LMI conditions nor the plant model (including how the input delay enters the state equation) used to obtain the controller.

    Authors: The LMI conditions are drawn from existing results on delayed state-feedback; however, the manuscript does not reproduce them or restate the precise plant model. We will insert the explicit LMI formulation together with the system equations (showing the input-delay term) to make the synthesis step self-contained. revision: yes

  3. Referee: [Output-feedback extension] Extension to output controllers: the claim that the same compensator-plus-observer architecture works for dynamic output feedback likewise lacks any stability argument or LMI that accounts for the simultaneous presence of both delays in the closed-loop map.

    Authors: The same limitation applies to the output-feedback extension. We will augment the revised manuscript with the corresponding composite Lyapunov analysis and LMI conditions that incorporate both mismatched delays in the dynamic output-feedback setting. revision: yes

Circularity Check

1 steps flagged

Central claim of direct combination for mismatched delays rests on self-cited compensators without new composite analysis

specific steps
  1. self citation load bearing [Abstract]
    "Second, this controller is realized using novel time-delay compensators \cite{trinhnam26}. This architecture successfully accommodates an output measurement delay τ_y that is independent of the input delay τ_u, enabling direct estimation of the delayed state-feedback control law."

    The claim that the architecture successfully handles independent delays is presented as following from the novel compensators in the self-cited prior work. The paper provides no independent verification or composite stability condition for combining the LMI controller, observer, and compensators; the central result therefore inherits its justification from the overlapping-author citation.

full rationale

The paper's two-stage architecture (LMI-synthesized delayed state-feedback followed by realization via compensators) asserts successful accommodation of independent τ_y and τ_u. This assertion is justified solely by citation to trinhnam26 for the compensators, with authors overlapping the present work. No new Lyapunov or LMI analysis is indicated for the observer error dynamics interacting with the compensators under mismatched delays, so the load-bearing integration step reduces to the prior self-cited result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; the ledger is therefore populated from the high-level statements that must hold for the claimed architecture to function.

axioms (2)
  • domain assumption The plant is a finite-dimensional linear time-invariant system whose state-feedback stabilization problem with input delay is solvable by existing LMI methods.
    Invoked when the first stage synthesizes the delayed state-feedback controller.
  • ad hoc to paper The time-delay compensators introduced in the cited reference can be combined with the LMI controller without introducing new instability.
    Required for the second stage that realizes the controller from delayed output measurements.

pith-pipeline@v0.9.1-grok · 5651 in / 1422 out tokens · 24013 ms · 2026-06-28T09:21:28.082153+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

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

  1. Delayed Functional Observers for Output-Delayed Linear Systems

    eess.SY 2026-06 unverdicted novelty 3.0

    Proposes a class of delayed functional observers for output-delayed linear systems to enable reconstruction of delayed control laws.

  2. 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

8 extracted references · 2 canonical work pages · cited by 2 Pith papers · 1 internal anchor

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

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

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