A constructive Lyapunov function design method for a class of homogeneous systems

In this paper a method to design homogeneous Lyapunov functions for a class of homogeneous dynamical systems is proposed. The method is constructive and simple. The procedure exploits homogeneity and polynomial-like properties of the class of functions considered for the vector fields and for the Lyapunov function candidates. It also makes strong use of an algebraic result known as Pólya´s Theorem to establish the required positive and negative definite properties of the Lyapunov function and its derivative. We illustrate the method by designing smooth Lyapunov functions for two systems. The first one is a Second Order Sliding Mode algorithm known as Super Twisting. We propose here for the first time a smooth Lyapunov function for it. The second example corresponds to a double integrator controlled by a continuous homogeneous state feedback, whose origin is finite time stable.