arxiv: 2604.08721 · v1 · submitted 2026-04-09 · 📡 eess.SY · cs.SY

Linear Feedback Controller for Homogeneous Polynomial Systems

Shaoxuan Cui , Qi Zhao , Guanlin Li , Hildeberto Jardon Kojakhmetov , Ming Cao This is my paper

Pith reviewed 2026-05-10 17:01 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords homogeneous polynomial systemsODECO tensorslinear feedback controlregion of attractionstabilizationinput-to-state stability

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The pith

For homogeneous polynomial systems whose nonlinearity is an ODECO tensor, a linear feedback sharing that tensor's eigenbasis supplies closed-form trajectories and sharp region-of-attraction estimates.

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

The paper seeks a stabilization method for homogeneous polynomial dynamical systems that avoids the conservatism and computational cost of local linearization or sum-of-squares techniques. It shows that when the nonlinear term admits an orthogonally decomposable tensor representation, a linear feedback can be chosen to share the same eigenbasis, preserving the structure in closed loop. This structure preservation produces explicit trajectory formulas, exact thresholds separating convergence from escape, and non-conservative characterizations of the basin of attraction. The same design also supplies explicit input-to-state stability bounds when bounded disturbances are present and mild gain conditions hold.

Core claim

For homogeneous polynomial systems whose nonlinear term is given by an orthogonally decomposable (ODECO) tensor, there exists a linear feedback gain that shares the ODECO eigenbasis. In the shared eigenbasis coordinates the closed-loop vector field decouples into independent scalar homogeneous equations whose solutions are known in closed form; from these solutions one obtains explicit convergence and escape time thresholds together with the precise region of attraction of the origin. When the feedback gains satisfy additional mild inequalities, the closed-loop system remains input-to-state stable with respect to bounded disturbances, again with explicit bounds.

What carries the argument

The structure-preserving linear feedback gain matrix chosen to share the eigenbasis of the system's orthogonally decomposable (ODECO) tensor, which keeps the closed-loop dynamics in the same coordinate system where the tensor is diagonal.

If this is right

Closed-loop trajectories admit explicit closed-form expressions once written in the shared eigenbasis coordinates.
Convergence and escape thresholds become computable directly from the eigenvalues and the feedback gains without optimization.
The region of attraction of the origin is characterized sharply and without the conservatism introduced by relaxations or local approximations.
Input-to-state stability bounds with respect to bounded disturbances hold whenever the feedback gains meet the stated mild inequalities.

Where Pith is reading between the lines

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

The method could serve as an exact benchmark against which more general but approximate design techniques are compared.
Systems whose tensors are only approximately ODECO might still use the nearest exact ODECO system as a nominal design and then add robustness margins.
The same eigenbasis-sharing idea may extend to time-varying or switched homogeneous polynomials whose instantaneous tensors remain ODECO.

Load-bearing premise

The nonlinear term must admit an orthogonally decomposable tensor representation, and a linear feedback gain must exist that shares the eigenbasis of that tensor while satisfying the mild conditions needed for the robustness bounds.

What would settle it

A numerical integration of the closed-loop ODE under the proposed feedback in which the observed state trajectory deviates from the explicit scalar solutions written in the shared eigenbasis coordinates, or an initial condition inside the claimed region of attraction that diverges.

Figures

Figures reproduced from arXiv: 2604.08721 by Guanlin Li, Hildeberto Jardon Kojakhmetov, Ming Cao, Qi Zhao, Shaoxuan Cui.

**Figure 1.** Figure 1: shows the basin classification on a grid and the boundary lines v ⊤ 1 x = ±1, which is indeed in line with Theorem 2. 2) ISS-type validation under matched disturbances (even p): We consider the matched disturbance model in (14) with d(t) = d1(t)v1 + d2(t)v2 and choose d1(t) = ¯d sin(ωt), d2(t) = ¯d cos(0.9ωt) with ¯d = 0.15 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: Even-p disturbed case ( ¯d = 0.15): modal time series y1(t), y2(t) with the predicted ultimate bounds (dashed). ¯d and compares measured ultimate magnitudes against the predicted curves. V. CONCLUSION This paper proposed a structure-preserving linear feedback design for homogeneous ODECO polynomial systems by enforcing that the controller shares the ODECO eigenbasis of the system tensor. This shared-basi… view at source ↗

**Figure 3.** Figure 3: Sweep over ¯d: measured ultimate magnitudes (markers) versus predicted bounds (dashed) from Theorem 4. [2] S. Cui, Q. Zhao, G. Zhang, H. Jardon-Kojakhmetov, and M. Cao, “Analysis of higher-order lotka-volterra models: Application of stensors and the polynomial complementarity problem,” IEEE Transactions on Automatic Control, 2025. [3] D. Angeli, “A tutorial on chemical reaction networks dynamics,” in 20… view at source ↗

read the original abstract

This paper studies stabilization and its corresponding closed-loop region-of-attraction (ROA) for homogeneous polynomial dynamical systems whose nonlinear term admits an orthogonally decomposable (ODECO) tensor representation. While recent tensor-based results provide explicit solutions and sharp global characterizations for open-loop ODECO systems, closed-loop synthesis and computable ROA estimates are still often dominated by local linearization or Lyapunov/SOS (sum of squares) methods, which can be conservative and computationally demanding. We propose a structure-preserving linear feedback design that shares the ODECO eigenbasis of the system's tensor, thereby enabling closed-form trajectory expressions, explicit convergence/escape thresholds, and sharp ROA characterizations. Under mild conditions, we further derive robustness/ISS-type bounds for bounded disturbances. Numerical examples validate the theoretical results.

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

For ODECO homogeneous polynomial systems a linear feedback can be chosen to share the tensor eigenbasis and deliver explicit closed-loop trajectories plus sharp ROA bounds.

read the letter

The main takeaway is that the authors give a linear controller for this restricted class of systems that keeps the closed-loop dynamics in a form where you can still write down the solutions in closed form and get non-conservative region-of-attraction estimates. They start from existing open-loop tensor results on ODECO systems and add a structure-preserving feedback that aligns with the eigenbasis, which lets them derive explicit convergence and escape thresholds directly. Under some mild extra conditions they also obtain ISS-style robustness bounds for bounded disturbances. Numerical examples are mentioned to back this up. That is a clean extension of the open-loop work and avoids the conservatism that usually comes with linearization or SOS methods. The approach is narrow by design: it only applies when the nonlinear term admits an ODECO decomposition and when a linear gain can be found that respects the eigenbasis. Within those limits the claims look sharp rather than loose. The main soft spot is that the abstract and stress-test give no derivation details or explicit conditions for when such a feedback exists, so it is still unclear how often the mild conditions hold in practice or how much extra work is needed to compute the gain. The citation pattern appears to rest on the prior tensor literature, which is appropriate. This paper is aimed at control researchers who already use tensor or polynomial models and want analytical rather than numerical tools for stability analysis. A reader who needs explicit ROA or trajectory formulas for homogeneous systems that happen to be ODECO would find it useful. I would send it to peer review; the claims are specific enough that referees can verify the derivations and check the examples against the stated assumptions.

Referee Report

0 major / 3 minor

Summary. The paper studies stabilization and region-of-attraction (ROA) estimation for homogeneous polynomial systems whose nonlinear term has an orthogonally decomposable (ODECO) tensor representation. It proposes a linear feedback controller designed to share the ODECO eigenbasis of the system tensor. This structure-preserving choice yields closed-form trajectory solutions, explicit convergence and escape time thresholds, sharp (non-conservative) ROA characterizations, and input-to-state stability (ISS) robustness bounds for bounded disturbances under stated mild conditions. The theoretical claims are illustrated with numerical examples.

Significance. If the derivations hold, the work supplies explicit, non-conservative closed-form results and sharp ROA/ISS bounds for a nontrivial subclass of polynomial systems. This is a clear improvement over the conservative local-linearization or SOS/Lyapunov estimates that dominate the literature for such systems. The explicit trajectory formulas and threshold conditions are particularly useful for analysis and could support practical controller synthesis in applications involving homogeneous polynomial dynamics.

minor comments (3)

[Abstract / §1] The abstract and introduction refer to 'mild conditions' for the ISS bounds without immediately stating them; these conditions should be listed explicitly (or cross-referenced to the relevant theorem) already in the introduction so readers can assess applicability at a glance.
[§5] Numerical examples are mentioned as validation but the manuscript would be strengthened by including a brief comparison (e.g., a table) of the obtained ROA volumes or convergence times against a standard SOS or linearization baseline on the same examples.
[§3] Notation for the eigenbasis-sharing property and the resulting closed-loop tensor is introduced in §3; a short remark clarifying that the feedback remains strictly linear (no state-dependent gain) would help readers unfamiliar with tensor decompositions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript on structure-preserving linear feedback control for ODECO homogeneous polynomial systems. The recommendation for minor revision is noted, and we appreciate the recognition of the explicit trajectory formulas, sharp ROA characterizations, and ISS bounds as improvements over conservative methods in the literature.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper assumes an ODECO tensor representation for the homogeneous polynomial nonlinearity as a starting point and proposes a linear feedback explicitly constructed to share that eigenbasis. This is a design choice under the given assumption rather than a self-referential definition or fitted prediction. No load-bearing self-citations, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation are indicated in the abstract or description. Closed-form trajectories and ROA characterizations follow directly from the shared eigenbasis under the ODECO structure, with numerical examples cited for validation. The derivation chain remains self-contained against the stated assumptions and does not reduce any claimed result to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the system tensor is orthogonally decomposable and that a linear feedback can be constructed to share its eigenbasis. No free parameters or new postulated entities are mentioned in the abstract.

axioms (1)

domain assumption The nonlinear term admits an orthogonally decomposable (ODECO) tensor representation.
This property is invoked as the enabling structure for the feedback design and closed-form analysis.

pith-pipeline@v0.9.0 · 5437 in / 1515 out tokens · 75430 ms · 2026-05-10T17:01:03.612282+00:00 · methodology

discussion (0)

Reference graph

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Hence, on the interval [0, Tε,r)we have|y r(t)|> ε >0and the transformation below is well-defined

Along trajectories starting in the ROA, the sign ofy r(t)is preserved andy r(t)→0(Theorem 2). Hence, on the interval [0, Tε,r)we have|y r(t)|> ε >0and the transformation below is well-defined. Lets r := sign(y r,0)∈ {−1,1}and defineu r(t) :=|y r(t)|−p >0.Using d dt |yr|=s r ˙yr (since sign(yr(t)) =s r on the ROA trajectory), we obtain ˙ur =−p|y r|−p−1 d d...

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