Final Set Adjustment in Barrier Function Adaptation Exploiting Properties of Signed Power-Based Controllers

Barrier function adaptation for sliding-mode controllers ensures not only that the system trajectories belong to the barrier function width but also the solutions, after a finite time, belong to a smaller positively invariant subset of the barrier function width whose size depends on the upper bound of the perturbations. In this article, the barrier function methodology is extended to a class of signed power-based controllers. It turns out that, whenever the upper bound of the perturbation is smaller than one, powers smaller than zero produce a smaller size of the positively invariant final set, and when the upper bound of the perturbation is bigger than one, the size of the final set for the same controllers is bigger. Finally, we propose a bipowered extension of the methodology to adjust the size of the final set for any value of the upper bound of the perturbation.