A Lyapunov Function for an Extended Super-Twisting Algorithm

Recently, an extension of the super-twisting algorithm for relative degrees
$m\geq {}1$
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.