A Lyapunov Function for an Extended Super-Twisting Algorithm

Recently, an extension of the super-twisting algorithm for relative degrees m >= 1 has been proposed. However, as of yet, no Lyapunov functions for this algorithm exist. This paper discusses the construction of Lyapunov functions by means of the sum-of-squares technique for m = 1. Sign definiteness of both Lyapunov function and its time derivative is shown in spite of numerically obtained-and hence possibly inexact-sum-of-squares decompositions. By choosing the Lyapunov function to be a positive semidefinite, the finite time attractivity of the system's multiple equilibria is shown. A simple modification of this semidefinite function yields a positive definite Lyapunov function as well. Based on this approach, a set of the algorithm's tuning parameters ensuring finite-time convergence and stability in the presence of bounded uncertainties is proposed. Finally, a generalization of the approach for m > 1 is outlined.