Strict Lyapunov Functions for the Super-Twisting Algorithm

A method to construct a family of strict Lyapunov functions, i.e., with negative definite derivative, for the super-twisting algorithm, without or with perturbations, is provided. This second order sliding modes algorithm is widely used to design controllers, observers and exact differentiators. The proposed Lyapunov functions ascertain finite time convergence, provide an estimate of the convergence time, and ensure the robustness of the finite-time or ultimate boundedness for a class of perturbations wider than the classical ones for this algorithm. Since the Lyapunov functions and their derivatives are quadratic forms, the operation with them is as simple as for linear time invariant systems.

I. Introduction

The Super-Twisting Algorithm (STA) is a well-known second order sliding modes (SOSM) algorithm introduced in [9], and it has been widely used for control [1], [4], [7], [9], [11], [12], observation [5], and robust exact differentiation [10], [15]. The STA can be written as

$$\eqalignno{{\mathdot {x}}_{1}= &\, -k_{1}\left \vert x_{1}\right \vert ^{1/2} {\rm sign}\left (x_{1}\right) +x_{2}+\varrho _{1}\left (x,t\right) \cr {\mathdot {x}}_{2}= &\, -k_{2} {\rm sign}\left (x_{1}\right) +\varrho _{2}\left (x,t\right) &{\hbox{(1)}}}$$ where are the scalar state variables, are gains to be designed, and are the perturbation terms. Under some conditions on , (1) is robust against a bounded perturbation , . Since the righthand side of (1) is discontinuous, its solutions will be understood in the sense of Filippov [6].