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arxiv: 2504.00165 · v3 · submitted 2025-03-31 · 🧮 math.OC · cs.SY· eess.SY

Robust Control of General Linear Delay Systems under Dissipativity: Part I -- A KSD-based Framework

Qian Feng , Wei Xing Zheng , Xiaoyu Wang , Feng Xiao This is my paper

Pith reviewed 2026-05-22 21:31 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords delay systemsdistributed delaysKrasovskii functionaldissipative controlKronecker-Seuret Decompositionrobust controllinear systemsintegral inequality
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The pith

The Kronecker-Seuret Decomposition factors or approximates L2 kernels of any number of distributed delays without conservatism to build complete Krasovskii functionals for dissipative controller design.

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

This paper develops a framework for memoryless dissipative full-state feedback control of linear systems that contain arbitrary finite numbers of pointwise and distributed delays in the state, input, and output. It applies the Kronecker-Seuret Decomposition to factorize or least-squares approximate the L2 kernels of those distributed delays from any number of such delays. The decomposition supports a complete-type Krasovskii functional whose integral kernels are made flexible by a new integral inequality obtained from the least-squares principle. Two theorems and an iterative algorithm then produce the controller gains without nonlinear solvers. A numerical example confirms the procedure works on a concrete system.

Core claim

The KSD enables factorization or least-squares approximation of any number of L2 DD kernels from any number of DDs without introducing conservatism. This facilitates the construction of a complete-type KF with flexible integral kernels by means of a novel integral inequality derived from the least-squares principle. Our solution includes two theorems and an iterative algorithm to compute controller gains without relying on nonlinear solvers.

What carries the argument

Kronecker-Seuret Decomposition (KSD), which performs factorization or least-squares approximation of matrix-valued L2 DD kernels from any number of distributed delays without conservatism.

If this is right

  • Memoryless dissipative full-state feedback becomes available for general linear delay systems with delays in state, input, and output.
  • Complete-type Krasovskii functionals can be constructed with flexible integral kernels.
  • Controller gains are obtained by an iterative algorithm that avoids nonlinear solvers.
  • The framework covers arbitrary finite numbers of pointwise and general distributed delays.

Where Pith is reading between the lines

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

  • The same decomposition and inequality could be tested on systems whose delays vary with time to check whether the no-conservatism property survives.
  • The iterative algorithm might be compared directly with discretization-based methods on high-dimensional examples to quantify any reduction in computational cost.
  • The dissipativity framework could be specialized to H-infinity or passivity cases by simple substitution of the supply rate without altering the KSD step.

Load-bearing premise

The Kronecker-Seuret Decomposition applies to arbitrary finite numbers of pointwise and general distributed delays while producing no conservatism, and the novel integral inequality from the least-squares principle holds for the complete-type functional.

What would settle it

A concrete linear delay system with distributed delays in which the KSD approximation forces conservatism that blocks existence of a feasible dissipative controller, or where the iterative algorithm fails to return gains despite a controller existing.

Figures

Figures reproduced from arXiv: 2504.00165 by Feng Xiao, Qian Feng, Wei Xing Zheng, Xiaoyu Wang.

Figure 2
Figure 2. Figure 2: Plots of u(t) = Kx(t) ensuring min γ = 0.6361 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plots of z(t) in (22) ensuring min γ = 0.6361 general LTDSs expressed via the Lebesgue-Stieltjes integrals, where the FDEs are understood in the extended sense. It has been shown that the KSD concept can overcome the challenges posed by the infinite dimension of DD-matrix kernels while ensuring that the synthesis conditions in Theo￾rem 1, Theorem 2 are represented by matrix inequalities with finite dimensi… view at source ↗
Figure 1
Figure 1. Figure 1: Plots of x(t) with K in Table II ensuring min γ = 0.6361 V. CONCLUSION We have proposed an SDP framework based on the KSD concept developed in [28] that can effectively address the dissipative controller design problem for the LTDS in (1) with general delay structures. The generality of the model in (1) is guaranteed as it closely mirrors the expressions of [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
read the original abstract

This paper introduces an effective framework for designing memoryless dissipative full-state feedback for general linear delay systems via the Krasovski\u{i} functional (KF) approach, where an arbitrary finite number of pointwise and general distributed delays (DDs) exists in the state, input and output. To handle the infinite dimensionality of DDs, we employ the Kronecker-Seuret Decomposition (KSD) which we recently proposed for analyzing matrix-valued functions in the context of delay systems. The KSD enables factorization or least-squares approximation of any number of $\fL^2$ DD kernels from any number of DDs without introducing conservatism. This also facilitates the construction of a complete-type KF with flexible integral kernels by means of a novel integral inequality derived from the least-squares principle. Our solution includes two theorems and an iterative algorithm to compute controller gains without relying on nonlinear solvers. A numerical example is tested to show the effectiveness of the proposed approach.

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 / 3 minor

Summary. The manuscript introduces a KSD-based framework for designing memoryless dissipative full-state feedback controllers for general linear delay systems containing an arbitrary finite number of pointwise and distributed delays in the state, input, and output. It employs the authors' prior Kronecker-Seuret Decomposition to factorize or least-squares approximate any number of L² DD kernels without conservatism, constructs a complete-type Krasovskii functional via a novel integral inequality derived from the least-squares principle, states two theorems, and provides an iterative algorithm for computing controller gains that avoids nonlinear solvers. Effectiveness is illustrated by a numerical example.

Significance. If the no-conservatism claim and the validity of the new integral inequality hold, the work offers a meaningful contribution to delay-system control by extending complete-type Lyapunov-Krasovskii methods to general distributed delays while bypassing both conservatism and nonlinear optimization. The iterative algorithm and explicit handling of delays in input/output channels distinguish it from prior approaches that often require restrictive assumptions on delay structure.

minor comments (3)
  1. The abstract states that the KSD 'enables factorization or least-squares approximation ... without introducing conservatism' but does not indicate where in the manuscript the precise statement of this property (including the finite-number and L² conditions) is proved or referenced; a forward pointer to the relevant theorem or lemma would improve readability.
  2. Notation for the distributed-delay kernels (e.g., the distinction between pointwise and general L² kernels in state/input/output channels) is introduced in the abstract but would benefit from an early dedicated notation table or subsection to avoid repeated unpacking in later sections.
  3. The numerical example is described as confirming the 'no-nonlinear-solver' claim, yet the manuscript does not report iteration counts, convergence tolerance, or a direct comparison against a standard LMI solver; adding these quantitative details would strengthen the practical-utility argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation for minor revision. The summary accurately captures the contributions of the KSD-based framework for dissipative control of general linear delay systems.

Circularity Check

0 steps flagged

Minor self-citation of prior KSD work; new results independent

full rationale

The paper cites its own prior work on the Kronecker-Seuret Decomposition (KSD) to handle arbitrary finite numbers of pointwise and distributed delays, but this is presented as an established tool rather than a load-bearing step that defines the target result. The central novelties—the novel integral inequality derived from the least-squares principle, the two theorems, and the iterative algorithm for controller gains—are developed independently in the present manuscript and do not reduce by construction to the KSD definition or to any fitted parameter renamed as a prediction. No self-definitional, uniqueness-imported, or ansatz-smuggled steps appear in the derivation chain; the self-citation is therefore minor and does not raise the circularity score above 2.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the framework rests on the domain assumption that the system is linear with finite delays and that KSD applies without conservatism; no free parameters or invented entities are explicitly introduced in the provided text.

axioms (2)
  • domain assumption The plant is a general linear system with an arbitrary finite number of pointwise and distributed delays in state, input, and output.
    Stated directly in the abstract as the class of systems addressed.
  • ad hoc to paper KSD factorization or least-squares approximation introduces no conservatism for any number of L2 DD kernels.
    Central to the no-conservatism claim and the construction of the KF.

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