Linear Feedback Controller for Homogeneous Polynomial Systems

TL;DR

This paper introduces a structure-preserving linear feedback controller for homogeneous polynomial systems with ODECO tensor structure, enabling explicit ROA estimates and robust stability analysis.

Contribution

It proposes a novel linear feedback design that preserves the ODECO tensor structure, allowing for explicit solutions and less conservative ROA estimates.

Findings

Explicit closed-form trajectories and ROA estimates are derived.

The method provides robustness bounds under bounded disturbances.

Numerical examples confirm the theoretical advantages.

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.