Construction of Lyapunov Functions for a Class of Higher Order Sliding Modes algorithms

A method to construct (strict) Lyapunov Functions for a class of Higher Order Sliding Modes (HOSM) algorithms, that are homogeneous and piecewise state affine is presented. It is shown first that several HOSM algorithms presented in the literature posses these properties. The basic idea of the construction method is borrowed from the constructive proofs of the Lyapunov´s Converse Theorems. It is shown, by means of some concrete examples of second and third order, that the construction of the Lyapunov Function can be done for this class of systems. The obtained Lyapunov functions allow the estimation of the convergence time, the values of the gains that render the origin finite time stable, and the robustness of the algorithms to bounded perturbations.