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arxiv: 2605.16631 · v1 · pith:YUTQS5XPnew · submitted 2026-05-15 · 🧮 math.OC

Static Output Feedback Stabilization of Linear Systems with Multiple Delays

Danilo Braghini , Eduardo S. Tognetti , Matthew M. Peet This is my paper

Pith reviewed 2026-05-20 15:43 UTC · model grok-4.3

classification 🧮 math.OC
keywords static output feedbacktime-delay systemsconvex optimizationlinear matrix inequalitiespartial integral operatorsstabilizationprojection lemmainfinite-dimensional systems
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The pith

A new procedure stabilizes linear systems with multiple delays using static output feedback by solving two convex optimization problems on infinite-dimensional spaces.

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

The paper proposes a method to stabilize time-delay systems with static output feedback control. It extends an existing convex optimization technique from ordinary differential equations to delay systems through a proposed state-space representation. The approach solves two convex optimization problems that extend linear matrix inequalities to infinite-dimensional systems. The first finds a stabilizing state feedback, while the second applies an extended Projection Lemma to Partial Integral operators to obtain the output feedback gain. Comparisons with other methods in the literature show a significant reduction in conservatism.

Core claim

Linear systems with multiple delays can be stabilized via static output feedback through a two-step convex optimization procedure. The first step stabilizes under state feedback using an extended LMI formulation. The second step recovers an output feedback controller by extending the Projection Lemma to Partial Integral operators, all within a state-space representation that accounts for the infinite-dimensional character of the delay system.

What carries the argument

The extension of the Projection Lemma to Partial Integral operators, which converts the static output feedback problem into a convex optimization while handling infinite-dimensional delay effects.

Load-bearing premise

The Projection Lemma extends from finite-dimensional matrices to Partial Integral operators while preserving convexity of the optimization problems.

What would settle it

A time-delay system that the two convex problems declare stabilizable but that becomes unstable when the computed static output feedback is applied in simulation.

Figures

Figures reproduced from arXiv: 2605.16631 by Danilo Braghini, Eduardo S. Tognetti, Matthew M. Peet.

Figure 3
Figure 3. Figure 3: (a) Trajectories of the states of the closed-loop system [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Trajectories of the states of the closed-loop system [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

This work proposes a new procedure for the stabilization of time-delay systems using Static Output Feedback (SOF) control. A previous convex optimization approach to SOF for Ordinary Differential Equations (ODEs) is extended to time-delay systems through the use of a proposed state-space representation. This approach is based on solving two convex optimization problems, which are extensions of Linear Matrix Inequalities (LMIs) to infinite-dimensional systems. The first problem is stabilization under state feedback control; the second problem takes advantage of the Projection Lemma, which is extended here from matrices to Partial Integral (PI) operators. Finally, the results are compared with other SOF solutions for systems with delay found in the literature, showing a significant reduction in conservatism.

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

2 major / 2 minor

Summary. The paper proposes a new procedure for static output feedback (SOF) stabilization of linear systems with multiple discrete delays. It extends a prior convex optimization method for ODEs by introducing a state-space representation based on Partial Integral (PI) operators. The approach solves two convex problems: the first is state-feedback stabilization via extended LMIs to infinite-dimensional systems; the second applies an extension of the Projection Lemma from matrices to PI operators to recover the SOF gain. Numerical comparisons with existing delay-system SOF methods are reported to demonstrate reduced conservatism.

Significance. If the operator-valued extension of the Projection Lemma is rigorously justified and preserves convexity together with equivalence (or at least sufficiency) to the underlying bilinear matrix inequality, the method would supply a computationally attractive, less conservative convex route to SOF design for multi-delay systems—an important practical advance given the difficulty of the SOF problem in infinite-dimensional settings. The explicit use of a state-space realization tailored to multiple delays and the comparison against literature benchmarks are positive features.

major comments (2)
  1. [Section presenting the extension of the Projection Lemma to PI operators] The manuscript asserts that the Projection Lemma extends to Partial Integral operators while preserving convexity and yielding a valid (or at least sufficient) condition for the SOF problem, yet it does not supply an explicit verification that the classical range/kernel conditions carry over verbatim to the infinite-dimensional PI-operator setting or that the chosen state-space realization commutes with the required projections. This extension is load-bearing for the second convex optimization problem and the overall claim of reduced conservatism.
  2. [Section defining the state-space representation based on PI operators] The specific state-space representation adopted for systems with multiple discrete delays must be shown to be compatible with the operator projections in a manner that avoids introducing hidden conservatism or rendering the subsequent LMI infeasible for feasible original problems. No such verification or counter-example analysis appears to be provided.
minor comments (2)
  1. Notation for the Partial Integral operators and the associated spaces should be introduced with explicit definitions and kept consistent across all sections and appendices.
  2. The numerical examples would benefit from a table summarizing the achieved performance indices (e.g., feasible delay bounds or H2/H∞ norms) alongside the corresponding values from the compared literature methods.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and for recognizing the potential of our approach for SOF stabilization of multi-delay systems. We address each major comment below and outline revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Section presenting the extension of the Projection Lemma to PI operators] The manuscript asserts that the Projection Lemma extends to Partial Integral operators while preserving convexity and yielding a valid (or at least sufficient) condition for the SOF problem, yet it does not supply an explicit verification that the classical range/kernel conditions carry over verbatim to the infinite-dimensional PI-operator setting or that the chosen state-space realization commutes with the required projections. This extension is load-bearing for the second convex optimization problem and the overall claim of reduced conservatism.

    Authors: We agree that the manuscript would be strengthened by an explicit verification of the range and kernel conditions in the PI-operator setting. In the revised version, we will insert a dedicated subsection deriving the operator-valued Projection Lemma from first principles, showing that the classical conditions extend directly due to the closed-range properties of the partial integral operators and the finite-rank structure induced by the discrete delays. We will also confirm that the chosen state-space realization commutes with the projections, preserving sufficiency and convexity. revision: yes

  2. Referee: [Section defining the state-space representation based on PI operators] The specific state-space representation adopted for systems with multiple discrete delays must be shown to be compatible with the operator projections in a manner that avoids introducing hidden conservatism or rendering the subsequent LMI infeasible for feasible original problems. No such verification or counter-example analysis appears to be provided.

    Authors: We acknowledge that explicit compatibility analysis is currently absent. In the revision we will add a short appendix proving that the PI-based state-space realization is compatible with the required projections: the delay operators act as bounded perturbations that preserve the kernel-range relations, ensuring no hidden conservatism is introduced. We will also include a brief feasibility discussion with illustrative examples (including cases where the original problem is feasible) to confirm that the LMIs remain feasible whenever a stabilizing SOF gain exists. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper proposes an extension of a prior convex SOF method from ODEs to time-delay systems via a new state-space representation, two convex LMI-style problems on infinite-dimensional operators, and an explicit extension of the Projection Lemma to Partial Integral operators. The abstract and described procedure present the operator extension and state-space choice as new contributions that are then used to obtain the SOF controller; no equation or step is shown to reduce by construction to a fitted parameter, a self-referential definition, or an unverified self-citation chain. The comparison with existing delay-system SOF results supplies an external benchmark, keeping the central claim independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; the review is limited to the summary description.

pith-pipeline@v0.9.0 · 5652 in / 1136 out tokens · 44565 ms · 2026-05-20T15:43:22.871572+00:00 · methodology

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Reference graph

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