A Lyapunov approach to second-order sliding mode controllers and observers

TLDR

The super‑twisting algorithm is widely used in sliding‑mode literature to design controllers, observers, and exact differentiators. This paper develops a strong Lyapunov function for the super‑twisting algorithm, a key second‑order sliding‑mode technique. The authors derive this Lyapunov function, providing a rigorous analytical tool for the algorithm. The Lyapunov function deepens understanding of finite‑time convergence and robustness, improves performance through linear corrections, enables handling of linearly growing perturbations, and opens new analysis and design tools for higher‑order sliding modes.

Abstract

In this paper a strong Lyapunov function is obtained, for the first time, for the super twisting algorithm, an important class of second order sliding modes (SOSM). This algorithm is widely used in the sliding modes literature to design controllers, observers and exact differentiators. The introduction of a Lyapunov function allows not only to study more deeply the known properties of finite time convergence and robustness to strong perturbations, but also to improve the performance by adding linear correction terms to the algorithm. These modification allows the system to deal with linearly growing perturbations, that are not endured by the basic super twisting algorithm. Moreover, the introduction of Lyapunov functions opens many new analysis and design tools to the higher order sliding modes research area.