Feedback Stabilization of Polynomial Systems: From Model-based to Data-driven Methods

TL;DR

The paper addresses global stabilization of input-affine polynomial systems by developing a model-based framework that removes the need for a radially unbounded Lyapunov function or constant $P$, using a generalized GAS condition and SOS feasibility. It extends this approach to data-driven stabilization from bounded-noise measurements, formulating a compatibility set $\\Sigma$ and employing a specialized S-lemma to obtain SOS conditions that guarantee GAS for all systems in $\\Sigma$, with optional incorporation of prior parameter knowledge to reduce conservatism. The methods are validated through illustrative examples showing global asymptotic stability of the closed-loop and tractable SOS formulations, highlighting practical applicability under model uncertainty and data noise. Overall, the work broadens the toolkit for robust, globally stabilizing controllers for polynomial systems, connecting model-based design with data-driven guarantees and offering pathways to real-world applications such as flexible-joint robotics.

Abstract

In this study, we propose new global stabilization approaches for a class of polynomial systems in both model-based and data-driven settings. The existing model-based approach guarantees global asymptotic stability of the closed-loop system only when the Lyapunov function is radially unbounded, which limits its applicability. To overcome this limitation, we develop a new global stabilization approach that allows a broader class of Lyapunov function candidates. Furthermore, we extend this approach to the data-driven setting, considering Lyapunov function candidates with the same functional structure. Using data corrupted by bounded noise, we derive conditions for constructing globally stabilizing controllers for unknown polynomial systems. Beyond handling noise, the proposed data-driven approach can be readily adapted to incorporate further prior knowledge of system parameters to reduce conservatism. In both approaches, sum-of-squares relaxation is used to ensure computational tractability of the involved conditions.

Figures (4)

• Figure 1: Phase portrait of the closed-loop system and the level sets of $V(x)$ in Example \ref{['ex 1']}. The gray arrows depict the closed-loop vector field and the black lines are trajectories starting from the edges and converging to the origin. The level sets are indicated by red dashed lines.
• Figure 2: Phase portrait of the closed-loop system and the level sets of $V(x)$ in Example \ref{['ex 1 datadriven']} (see \ref{['fig 2']}) and Example \ref{['ex 3']} (see \ref{['fig 3']}). The gray arrows depict the closed-loop vector field and the black lines are trajectories starting from the edges and converging to the origin. The level sets are indicated by red dashed lines.
• Figure 3: The mass-spring system with a hardening spring.
• Figure 4: Trajectories of the true closed-loop system for four different initial states. The zero state in each subplot is depicted by a red line.

Theorems & Definitions (13)

• Lemma 1
• proof
• Theorem 1
• proof
• Example 1
• Definition 1
• Lemma 2
• Theorem 2
• proof
• Remark 1