Lyapunov Stability Analysis of a Mass-Spring system subject to Friction

TL;DR

This paper presents a Lyapunov-based method for analyzing the stability of a mass-spring system with friction, addressing stick-slip behavior and providing conditions for global asymptotic stability.

Contribution

It introduces a numerically tractable approach combining attractor characterization and basin of attraction estimation for stability analysis.

Findings

Derived conditions ensure global asymptotic stability.

Validated approach with an illustrative example.

Provides a systematic way to analyze stick-slip dynamics.

Abstract

This paper deals with the stability analysis of a mass-spring system subject to friction using Lyapunov-based arguments. As the described system presents a stick-slip phenomenon, the mass may then periodically sticks to the ground. The objective consists of developing numerically tractable conditions ensuring the global asymptotic stability of the unique equilibrium point. The proposed approach merges two intermediate results: The first one relies on the characterization of an attractor around the origin, to which converges the closed-loop trajectories. The second result assesses the regional asymptotic stability of the equilibrium point by estimating its basin of attraction. The main result relies on conditions allowing to ensure that the attractor issued from the first result is included in the basin of attraction of the origin computed from the second result. An illustrative example draws the interest of the approach.