Lyapunov stability analysis of a mass–spring system subject to friction

• Published in: Systems & control letters Vol. 150; p. 104910
• Main Authors: Barreau, Matthieu, Tarbouriech, Sophie, Gouaisbaut, Frédéric
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.04.2021; Elsevier
• Subjects: Analysis of PDEs; Attractor; Friction; Global asymptotic stability; LMI; Lyapunov methods; Mathematics; Optimization and Control; Regional asymptotic stability; Global asymptotic stability; LMI; Friction; Regional asymptotic stability; Attractor; Lyapunov methods
• ISSN: 0167-6911; 1872-7956; 1872-7956
• DOI: 10.1016/j.sysconle.2021.104910

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 stick 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. •The stability analysis of mass–spring system subject to a nonlinear friction force is conducted using quadratic Lyapunov functions, leading to stability tests expressed by LMIs.•We give a precise estimation of a global attractor with emphasis on the practical consequences.•If there exists a basin of attraction, we provide an algorithm to give an inner-estimation of the latter.•We mix the two previous results to state the global exponential stability of the system. This work proves then rigorously what was experimentally already noted: for a reference speed large enough, the unique equilibrium point is globally exponentially stable.