Dissipative Control of General Linear Time-Delay Systems: Applications of the Kronecker-Seuret Decomposition

Feng Xiao; Qian Feng; Wei Xing Zheng; Xiaoyu Wang

arxiv: 2202.10397 · v12 · submitted 2022-02-21 · 🧮 math.OC

Dissipative Control of General Linear Time-Delay Systems: Applications of the Kronecker-Seuret Decomposition

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

Pith reviewed 2026-05-24 12:13 UTC · model grok-4.3

classification 🧮 math.OC

keywords Kronecker-Seuret decompositiontime-delay systemsdissipative controlLyapunov-Krasovskii functionalsdistributed delaysintegral inequalitieslinear systemscomplete-type functionals

0 comments

The pith

Kronecker-Seuret decomposition factorizes matrix kernels without conservatism to build complete Lyapunov functionals for dissipative control of linear systems with arbitrarily many delays.

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

The paper develops a unified framework for stabilizing general linear time-delay systems that include both pointwise and distributed delays, even when the number of each is unlimited. It applies the Kronecker-Seuret decomposition to factorize and approximate the matrix-valued integral kernels simultaneously. This factorization supports the construction of complete-type Lyapunov-Krasovskii functionals whose kernels can incorporate any number of weakly differentiable linearly independent functions. Novel least-squares integral inequalities derived from this approach close the stability conditions. The resulting synthesis yields two theorems and an iterative algorithm that compute controller gains directly, without nonlinear solvers, and is demonstrated on two previously intractable examples.

Core claim

The Kronecker-Seuret decomposition can factorize and approximate different kernel functions simultaneously without introducing conservatism, enabling construction of complete-type functionals whose integral kernels include any number of weakly differentiable linearly independent functions for dissipative control of systems with unlimited delays.

What carries the argument

The Kronecker-Seuret decomposition (KSD) of matrix-valued functions, which performs simultaneous factorization and approximation of multiple delay kernels.

If this is right

Two theorems plus an iterative algorithm supply the controller gains for each of the two dissipative control problems.
The same KSD-based construction applies uniformly to systems whose distributed-delay kernels contain any finite number of square-integrable functions.
Complete-type functionals can now incorporate any collection of weakly differentiable linearly independent kernel functions.
Numerical solution of the synthesis problems no longer requires nonlinear optimization routines.

Where Pith is reading between the lines

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

The same factorization technique might be applied to derive non-conservative bounds for other classes of integral functionals appearing in delay systems.
If the KSD can be computed for kernels arising in nonlinear or time-varying delay systems, the method could extend beyond the linear autonomous case treated here.
The iterative algorithm's convergence behavior under increasing numbers of kernel functions could be studied as a separate numerical question.

Load-bearing premise

The Kronecker-Seuret decomposition factors the matrix kernels of distributed delays without adding conservatism even when those kernels contain arbitrarily many square-integrable functions, and the derived least-squares inequalities remain valid for the resulting functionals.

What would settle it

A concrete linear system with distributed delays whose matrix kernel contains several linearly independent square-integrable functions, for which the KSD-based controller computed from the theorems fails to enforce dissipativity.

Figures

Figures reproduced from arXiv: 2202.10397 by Feng Xiao, Qian Feng, Wei Xing Zheng, Xiaoyu Wang.

**Figure 1.** Figure 1: The close-loop system’s trajectories x(t) 6.2. DSFC for an LTDS with Controllers Delays This subsection aims to show the advantages of adding delays to controllers when system (1) has no input delays. Consider system (1) with the same parameters as in subsection 6.1 except for the input delay matrices Bi = Bei(τ ) = On,p and Bi = Be i(τ ) = Om,n, ∀i ∈ Nν. In this case, we can employ the controller defined … view at source ↗

**Figure 2.** Figure 2: The trajectory of the control action u(t) [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗

**Figure 3.** Figure 3: The regulated output z(t) [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗

**Figure 4.** Figure 4: The Close-Loop System’s Trajectories x(t) with li ∈ N and λi ∈ N0 for all i ∈ Nν, which implies ∀i ∈ Nν, Fi = R K ϖ(τ )fi(τ )f ⊤ i (τ )dτ 0. Then ∀x(·) ∈ L2 ϖ(K; R n ), Xn i=1 Z Ki ϖ(τ )x ⊤(τ )Xix(τ )dτ ≥ [∗] ν diag i=1 [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗

**Figure 5.** Figure 5: Controller Action u(t) Since dim ϕ ⊤ i (·) φ⊤ i (·) f ⊤ i (·) ⊤ = dim (gi(τ )) = κi in (8)–(9) can be increased without limits, there always exist Ai,j ∈ R n×n, Ci,j ∈ R m×n, Bi,j ∈ R n×p , Bi,j ∈ R m×p and gi(τ ) = [gi,j (τ )]κi j=1 such that for all i ∈ Nν and τ ∈ Ii Aei(τ ) = Xκi j=1 Ai,jgi,j (τ ), Cei(τ ) = Xκi j=1 Ci,jgi,j (τ ), Bej (τ ) = Xκi j=1 Bi,jgi,j (τ ), Be i(τ ) = Xκi j=1 Bi,jgi,j (τ ) (B.1) … view at source ↗

**Figure 6.** Figure 6: The Regulated Output z(t) Banks, H. T., & Burns, J. A. 1978. Hereditary Control Problems: Numerical Methods Based on Averaging Approximations. SIAM Journal on Control and Optimization, 16(2), 169–208. Banks, H. T., Rosen, I. G., & Ito, K. 1984. Spline Based Technique for Computing Riccati Operators and Feedback Controls in Regulator Problems for Delay Equations. SIAM Journal on Scientific and Statistical C… view at source ↗

read the original abstract

Stabilizing autonomous linear time delay systems, particularly when addressing an unlimited number of pointwise and distributed delays (DDs) under dissipative constraints, poses a significant challenge. Existing solutions are often hindered by theoretical limitations, numerical obstacles, or an inability to address the complexities of the delay integral kernels. In this paper, we propose a unified framework to tackle the above problem by employing the concept of the Kronecker-Seuret decomposition (KSD) for matrix-valued functions, which we recently have developed for the analysis of complex delay structures in coordination with the Krasovski\u{\i} functional approach. Our strategy can simultaneously address two distinct control problems, where the matrix kernels of DDs can contain an unlimited number of square-integrable functions. We show in detail how the KSD can factorize and approximate different kernel functions simultaneously without introducing conservatism. Furthermore, the use of KSD also enables us to construct complete-type functionals, whose integral kernels can include any number of weakly differentiable and linearly independent functions, underpinned by the utilization of novel integral inequalities derived from the least-squares principle. The solution to each synthesis problem comprises two theorems accompanied by an iterative algorithm, which can be utilized as a single package to compute controller gains, thus eliminating the need for nonlinear solvers. We present the testing results of two challenging examples, which could not be addressed by existing methods, to demonstrate the effectiveness of our methodology. Additionally, the paper reviews recent advancements in the research of time-delay systems, providing a valuable reference for both emerging and established researchers.

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 gives a unified KSD-based framework for dissipative control of linear systems with unlimited pointwise and distributed delays, plus iterative synthesis algorithms, but the zero-conservatism guarantee for arbitrary kernels rests on the least-squares inequalities holding exactly for the constructed functionals.

read the letter

The main advance is taking the authors' prior Kronecker-Seuret decomposition and using it to factorize and approximate matrix kernels with any number of square-integrable functions at once, then building complete-type Krasovskii functionals whose integral terms can include arbitrarily many weakly differentiable linearly independent functions. This lets them state dissipativity conditions for two synthesis problems and supply iterative algorithms that compute gains without nonlinear solvers. The two numerical examples are ones prior methods could not handle, which is concrete evidence the approach reaches cases that were previously out of reach. The literature review section also serves as a compact reference for recent work on time-delay systems. The soft spot is exactly the load-bearing claim in the stress-test note: that KSD introduces no conservatism when handling unlimited kernels simultaneously and that the new least-squares inequalities apply to the resulting functionals without residual positive-semidefinite terms. The abstract asserts both, but the validity for growing numbers of functions is what determines whether the dissipativity LMI conditions and the iterative procedure remain non-conservative in practice. Without the full derivations and error bounds it is not possible to confirm the claim holds with equality in the limit. This work is aimed at control researchers who already work with distributed delays and complete-type functionals; anyone needing to synthesize controllers for systems with complex or high-dimensional delay kernels would find the unified treatment and the algorithm useful. It deserves a serious referee because the framework is new, the examples are non-trivial, and the central technical question (whether the inequalities close without remainder) is well-posed and falsifiable.

Referee Report

2 major / 2 minor

Summary. The paper claims to provide a unified framework for dissipative control of general linear time-delay systems with unlimited pointwise and distributed delays. It employs the Kronecker-Seuret decomposition (KSD) together with the Krasovskii functional approach to simultaneously address two control problems. The central assertions are that KSD enables simultaneous factorization and approximation of matrix kernels containing an arbitrary number of square-integrable functions without conservatism, that this permits construction of complete-type functionals whose kernels include any number of weakly differentiable linearly independent functions, and that novel least-squares integral inequalities support dissipativity conditions. Two theorems plus an iterative algorithm are given for computing controller gains without nonlinear solvers, with results shown on two challenging examples.

Significance. If the non-conservatism claim and validity of the inequalities for arbitrarily many functions hold, the work would advance dissipative control design for systems with complex, unlimited delay structures, a longstanding challenge. Credit is due for the explicit iterative algorithm package and for testing on examples stated to be beyond existing methods; the review of recent time-delay research also adds reference value.

major comments (2)

[Abstract and § on complete-type functionals] Abstract, paragraph on strategy and complete-type functionals: the claim that KSD factorizes/approximates arbitrary numbers of square-integrable kernel functions simultaneously with no conservatism is load-bearing for the dissipativity conditions and the iterative algorithm. The manuscript must demonstrate that the novel least-squares integral inequalities remain valid without positive semidefinite residual terms for the constructed complete-type functionals when the number of linearly independent functions grows without bound; otherwise the 'no conservatism' guarantee does not follow.
[Theorems 1 and 2] Theorems on controller synthesis: the reduction of the dissipativity LMI conditions to quantities obtained from the KSD-fitted parameters must be shown explicitly. If the least-squares inequalities only hold approximately for large numbers of functions, the iterative algorithm may still require post-hoc tuning that reintroduces conservatism, contradicting the central claim.

minor comments (2)

[Notation and preliminaries] Notation for the matrix-valued kernel functions should be made uniform between the KSD definition and the subsequent functional construction to avoid ambiguity when the number of basis functions increases.
[Numerical examples] The two numerical examples would benefit from tabulated controller gains, closed-loop eigenvalues or decay rates, and explicit comparison metrics against any prior methods that could partially address the examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive feedback on our manuscript. We address each major comment below with clarifications and indicate revisions where appropriate to strengthen the presentation of the no-conservatism claims.

read point-by-point responses

Referee: [Abstract and § on complete-type functionals] Abstract, paragraph on strategy and complete-type functionals: the claim that KSD factorizes/approximates arbitrary numbers of square-integrable kernel functions simultaneously with no conservatism is load-bearing for the dissipativity conditions and the iterative algorithm. The manuscript must demonstrate that the novel least-squares integral inequalities remain valid without positive semidefinite residual terms for the constructed complete-type functionals when the number of linearly independent functions grows without bound; otherwise the 'no conservatism' guarantee does not follow.

Authors: The KSD provides an exact algebraic factorization of any finite collection of square-integrable kernels into a Kronecker-structured sum, independent of the number of summands. The least-squares integral inequalities are obtained by projecting onto the finite-dimensional span of the chosen linearly independent functions; the resulting residual is identically zero within that span by construction of the projection. Consequently the inequalities hold exactly (no positive-semidefinite remainder) for every finite number of functions, and the complete-type functional is assembled directly from these exact bounds. We will add a short remark and one-line proof in the revised section on complete-type functionals confirming that the argument is dimension-independent and therefore remains valid as the number of functions increases without bound. revision: yes
Referee: [Theorems 1 and 2] Theorems on controller synthesis: the reduction of the dissipativity LMI conditions to quantities obtained from the KSD-fitted parameters must be shown explicitly. If the least-squares inequalities only hold approximately for large numbers of functions, the iterative algorithm may still require post-hoc tuning that reintroduces conservatism, contradicting the central claim.

Authors: The proofs of Theorems 1 and 2 substitute the KSD coefficients directly into the Krasovskii derivative, apply the exact least-squares inequalities derived above, and obtain a set of LMIs whose decision variables are precisely the KSD-fitted matrices. No further approximation or tuning step appears. The iterative algorithm simply solves these LMIs sequentially; because the inequalities are exact, the feasibility of the LMIs is equivalent to dissipativity of the closed-loop system. We will expand the proof paragraphs to display the substitution steps explicitly and will add a sentence stating that the algorithm requires no post-hoc adjustment. revision: yes

Circularity Check

0 steps flagged

No significant circularity; KSD application and synthesis rest on independent constructions shown in this work

full rationale

The paper cites its own prior development of the Kronecker-Seuret decomposition but explicitly demonstrates within this manuscript how KSD factorizes and approximates kernel functions simultaneously without conservatism, constructs complete-type functionals with any number of weakly differentiable functions, and derives novel least-squares integral inequalities. The two synthesis theorems and iterative algorithm for controller gains rely on these elements together with the Krasovskii functional approach rather than reducing by definition or construction to quantities defined in the prior citation. No step matches any enumerated circularity pattern; the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard background results in functional analysis and stability theory plus the authors' prior KSD construction; no explicit free parameters or new postulated entities are introduced in the abstract.

axioms (2)

domain assumption Krasovskii functional approach remains valid when combined with matrix-valued function decompositions for time-delay stability
Invoked when the paper states the strategy coordinates KSD with the Krasovskii functional approach.
ad hoc to paper Least-squares principle yields valid integral inequalities for bounding derivatives of complete-type functionals containing arbitrarily many linearly independent functions
Cited as the underpinning for the novel inequalities used to construct the functionals.

pith-pipeline@v0.9.0 · 5815 in / 1505 out tokens · 27572 ms · 2026-05-24T12:13:48.571791+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

complete

Springer. Feng, Q., Nguang, S. K., & Seuret, A. 2020. Stability Analysis of Linear Coupled Diﬀerential-Diﬀerence Systems with General Distributed Delays. IEEE Transactions on Automatic Control , 65(3), 1356–1363. Feng, Qian, & Nguang, Sing Kiong. 2016. Stabilization of Uncertain Linear Distributed Delay Systems with Dissipativity Constraints. Systems & Co...

work page arXiv 2020
[2]

36 Krasovskii, NN

Springer Science & Business Media. 36 Krasovskii, NN. 1963. Stability of Motion . Stanford University Press Stanford. Translation with additions of the 1959 Russian edition. Krstic, M. 2008. Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch. Automatica, 44(11), 2930–2935. Kunisch, Karl. 1982. App...

work page 1963

[1] [1]

complete

Springer. Feng, Q., Nguang, S. K., & Seuret, A. 2020. Stability Analysis of Linear Coupled Diﬀerential-Diﬀerence Systems with General Distributed Delays. IEEE Transactions on Automatic Control , 65(3), 1356–1363. Feng, Qian, & Nguang, Sing Kiong. 2016. Stabilization of Uncertain Linear Distributed Delay Systems with Dissipativity Constraints. Systems & Co...

work page arXiv 2020

[2] [2]

36 Krasovskii, NN

Springer Science & Business Media. 36 Krasovskii, NN. 1963. Stability of Motion . Stanford University Press Stanford. Translation with additions of the 1959 Russian edition. Krstic, M. 2008. Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch. Automatica, 44(11), 2930–2935. Kunisch, Karl. 1982. App...

work page 1963